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Bieri-Neumann-Strebel-Renz invariants and tropical varieties of integral homology jump loci

Yongqiang Liu Institute of Geometry and Physics, University of Science and Technology of China, Hefei 230026, P.R. China [email protected] Yuan Liu Institute of Geometry and Physics, University of Science and Technology of China, Hefei 230026, P.R. China [email protected]

(4 avril 2025)

Résumé

Papadima and Suciu studied the relationship between the Bieri-Neumann-Strebel-Renz (short as BNSR) invariants of spaces and the homology jump loci of rank one local systems. Recently, Suciu improved these results using the tropical variety associated to the homology jump loci of complex rank one local systems. In particular, the translated positive-dimensional component of homology jump loci can be detected by its tropical variety. In this paper, we generalize Suciu’s results to integral coefficients and give a better upper bound for the BNSR invariants. Then we provide applications mainly to Kähler groups. Specifically, we classify the Kähler group contained in a large class of groups, which we call the weighted right-angled Artin groups. This class of groups comes from the edge-weighted finite simple graphs and is a natural generalization of the right-angled Artin groups.

1 Introduction

1.1 Background

In 1987, a powerful group theoretic invariant was introduced by Bieri, Neumann and Strebel in [BNS87], now called the BNS invariant. This invariant is a generalization of a former invariant studied by Bieri and Strebel in [BS80, BS81] for metabelian groups. The BNS invariant was later generalized to higher degrees for groups by Bieri and Renz [BR88] and from groups to spaces by Farber, Geoghegan, and Schütts in [FGS10]. These invariants are called the Bieri-Neumann-Strebel-Renz (short as BNSR) invariants, which record the geometric finiteness properties of the spaces.

The computation of the BNSR invariant is extremely difficult. Even in degree $1$ case, it is only known for restricted types of groups, such as metabelian groups [BS80, BS81, BG84], one relator groups [Bro87], right-angled Artin groups [MV95, MMV98], Kähler groups [Del10] and pure braid groups [KMM15], etc. Papadima and Suciu in [PS10] initiated the project of looking for approximations of the BNSR invariants, which (1) are more computable and (2) are rationally defined upper bounds for the BNSR invariants. These bounds are derived from the homology jump loci, defined using the homology of the space with field coefficients in rank one local systems. Recently, Suciu improved this bound in [Suc21] using the tropical variety associated to the homology jump loci of rank one local systems with complex coefficients. In particular, the translated positive-dimensional component of homology jump loci can be detected by its tropical variety.

In this paper, we follow Suciu’s approach in [Suc21] and study the tropical varieties of homology jump loci with integral coefficients. The complement of these tropical varieties gives better upper bounds for the BNSR invariants.

1.2 Main results

Let $X$ be a connected finite CW complex with $\pi_{1}(X)=G$ . Let $\mathrm{S}(G)$ denote the unit sphere in the real vector space $\mathrm{Hom}(G;\mathbb{R})\cong H^{1}(X;\mathbb{R})$ . In this paper, we always assume that $\dim H^{1}(X;\mathbb{R})>0$ . Set $H=H_{1}(X;\mathbb{Z})$ , which is the abelianization of $G$ . Then it is clear that $\mathrm{S}(G)=\mathrm{S}(H)$ . We say $\chi\in\mathrm{S}(G)$ is rational if the image of $\chi$ is isomorphic to $\mathbb{Z}$ . For any integer $k\geqslant 0$ , the $k$ -th BNSR invariant $\Sigma^{k}(X;\mathbb{Z})$ (see Definition 4.2) forms a decreasing sequence of open subsets of $\mathrm{S}(G)$ as $k$ increases.

Let $\mathbbm{k}$ be a coefficient field. The homology jump ideal $\mathcal{J}^{\leqslant k}(X;\mathbb{Z})$ (resp. $\mathcal{J}^{\leqslant k}(X;\mathbbm{k})$ ) can be defined via the cellular chain complex of the maximal abelian cover of $X$ with coefficients in $\mathbb{Z}$ (resp. $\mathbbm{k}$ ), as a complex of $\mathbb{Z}H$ (resp. $\mathbbm{k}H$ ) modules (see Definition 4.7). In fact, $\mathcal{J}^{\leqslant k}(X;\mathbb{Z})$ (resp. $\mathcal{J}^{\leqslant k}(X;\mathbbm{k})$ ) is an ideal in $\mathbb{Z}H$ (resp. $\mathbbm{k}H$ ). When $\mathbbm{k}$ is an algebraically closed field, the variety of the ideal $\mathcal{J}^{\leqslant k}(X;\mathbbm{k})$ is exactly the homology jump loci $\mathcal{V}^{\leqslant k}(X;\mathbbm{k})$ , i.e, the collection of the rank one $\mathbbm{k}$ -coefficient local systems on $X$ such that its homology is non-zero for some degree in the range $[0,k]$ (see Definition 4.10). We refer the readers to Suciu’s survey paper [Suc11] for a comprehensive background on this topic.

For any ideal $I\subset\mathbb{Z}H$ (resp. $\mathbbm{k}H$ ), one can define its tropicalization ${\mathrm{Trop}}_{\mathbb{Z}}(I)$ (resp. ${\mathrm{Trop}}_{\mathbbm{k}}(I)$ ) in $\mathrm{Hom}(H;\mathbb{R})$ . Since tropical varieties over $\mathbb{Z}$ are relatively uncommon, we provide a detailed study in section 3. For any subset $Z\subseteq\mathrm{Hom}(H;\mathbb{R})$ , denote the image of $Z-\{0\}$ in $\mathrm{S}(H)$ under natural projection as $\mathrm{S}(Z)$ . Our main result reads as follows.

Theorem 1.1.

With the above notations and assumptions, we have

\Sigma^{k}(X;\mathbb{Z})\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{\leqslant k}(X;\mathbb{Z}))\big{)}^{c}\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{J}^{\leqslant k}(X;\mathbbm{k}))\big{)}^{c}

(1.1)

Moreover, $\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{\leqslant k}(X;\mathbb{Z}))\big{)}$ and $\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{J}^{\leqslant k}(X;\mathbbm{k}))\big{)}$ are both finite unions of rationally defined convex cones over polyhedrons on the sphere $\mathrm{S}(G)$ . In particular, they both have dense rational points.

Remark 1.2.

When $k=1$ , the first inclusion in eq. 1.1 is essentially due to Bieri, Groves and Stebel in [BS80, BS81, BG84]. Moreover, they showed that if $G$ is a finitely generated metabelian group, the first inclusion becomes an equality (for $k=1$ ). For more details, see section 4.4. On the other hand, the first inclusion in eq. 1.1 could be strict, see Example 5.2.

Theorem 1.1 is inspired by Suciu’s recent work [Suc21]. In particular, Theorem 1.1 recovers [Suc21, Theorem 1.1], which asserts that

\Sigma^{k}(X;\mathbb{Z})\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{C}}(\mathcal{J}^{\leqslant k}(X;\mathbb{C}))\big{)}^{c}.

See Remark 4.12 for more details. One can adapt Suciu’s proof to show that

\Sigma^{k}(X;\mathbb{Z})\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{J}^{\leqslant k}(X;\mathbbm{k}))\big{)}^{c}

for any algebraically closed field coefficients $\mathbbm{k}$ . In general the inclusion

\bigcup_{\mathrm{char}(\mathbbm{k})=p\geqslant 0}\mathrm{S}({\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{J}^{\leqslant k}(X;\mathbbm{k}))\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{\leqslant k}(X;\mathbb{Z}))\big{)}

could be strict, see Example 5.3. Following directions pointed out by Bieri and Groves in [BG84, Section 8.4], we show that the missing ingredient is the tropical variety for the $p$ -adic valuation over $\mathbb{Q}$ as in Proposition 3.9 (see Remark 3.10 for more explanations).

The proof of Theorem 1.1 replies on a series of nice results due to Bieri, Groves, and Strebel [BS80, BS81, BG84]. They gave a complete description for the Sigma-invariants of finitely generated modules over finitely generated abelian groups. Applying their results and a key theorem due to Papadima and Suciu [PS10, Theorem 10.1], we obtain Theorem 1.1. Since Bieri, Groves, and Strebel’s results are one of the origins of tropical geometry (see [EKL06]), one can translate the invariant they studied into the language of tropical geometry, and this is why the tropical variety shows up in Theorem 1.1.

1.3 Applications

It is a question of Serre to characterize finitely presented groups that can serve as the fundamental group of a compact Kähler manifold, called the Kähler groups. While some obstructions are known mainly due to the Hodge theory, we still do not have a panorama of this class of groups. The readers may refer to the monographs [ABC⁺96, Py24] and the survey papers [Ara95, Bur11] for this interesting topic.

A relative version of Serre’s question would be to describe the intersection of Kähler groups with another class of groups. To name a few non-trivial known cases, we have the classification of Kähler groups within $3$ -dimensional manifold groups in [DS09, Kot12b, BMS12]; within right-angled Artin groups in [DPS09]; within one-relator groups in [BM12]; within groups of large deficiency in [Kot12a]; within cubulable groups up to finite index in [DP19], etc. Under this spirit, we classify Kähler groups among a new class of groups, which is a natural generalization of the right-angled Artin groups. We call them the weighted right-angled Artin groups. This class of groups comes from the edge-weighted finite simple graphs.

Definition 1.3 (Weighted right-angled Artin groups).

Let $\Gamma_{\ell}=(V,E,\ell)$ be an edge-weighted finite simple graph, with vertex set $V$ , edge set $E$ and an edge weight function $\ell\colon E\to\mathbb{Z}_{>0}$ . The weighted right-angled Artin group associated to $\Gamma_{\ell}$ is the group $G_{\Gamma_{\ell}}$ generated by the vertices $a\in V$ , with a defining relation

[a_{i},a_{j}]^{\ell(e)}=1

for each edge $e=\{a_{i},a_{j}\}$ in $E$ (here $[a_{i},a_{j}]=a_{i}a_{j}a_{i}^{-1}a_{j}^{-1}$ ). If $\ell(e)=1$ for all $e\in E$ , then $G_{\Gamma_{\ell}}$ is the classical right-angled Artin group, denoted by $G_{\Gamma}$ .

Remark 1.4.

The weighted right-angled Artin groups are constructed in a way similar to Artin groups. Moreover, the following Coxeter group

\langle a_{i}\in V|a_{i}^{2}=1,(a_{i}a_{j})^{2\ell(e)}=1\text{ when there is an edge }e=\{a_{i},a_{j}\}\rangle

is a quotient of the weighted right-angled Artin group $G_{\Gamma_{\ell}}$ .

The various properties of the right-angled Artin group have been thoroughly studied by Papadima and Suciu in [PS06, PS09]. Moreover, the Kähler right-angled Artin group is classified by Dimca, Papadima, and Suciu as follows (the same result is proved by Py using different methods in [Py13, Corollary 4]).

Theorem 1.5 ([DPS09], Corollary 11.14).

Let $\Gamma$ be a finite simple graph and let $G_{\Gamma}$ denote the corresponding right-angled Artin group. Then the following are equivalent.

(i)

The group $G_{\Gamma}$ is Kähler.
(ii)

The graph $\Gamma$ is a complete graph on an even number of vertices.
(iii)

The group $G_{\Gamma}$ is a free abelian group of even rank.

We classify Kähler weighted right-angled Artin group as follows.

Theorem 1.6.

For a weighted right-angled Artin group $G_{\Gamma_{\ell}}$ , the following are equivalent.

(i)

The group $G_{\Gamma_{\ell}}$ is Kähler.
(ii)

The edge weighted graph $\Gamma_{\ell}$ is a complete graph on an even number of vertices and no edges with weight $\geqslant 2$ are adjacent.
(iii)

The group $G_{\Gamma_{\ell}}$ is a finite product of groups with type $\langle a_{1},a_{2}|[a_{1},a_{2}]^{m}=1\rangle$ for some positive integer $m$ .

Remark 1.7.

Professor Delzant kindly point out to us that the weighted right-angled Artin group $G_{\Gamma_{\ell}}$ is cubulable if the weights $\ell(e)\geqslant 2$ for all edges $e\in E$ . In this case, our result is compatible (up to finite index) with his work with Py in [DP19].

Dimca, Papadima and Suciu indeed classified quasi-Kähler right-angled Artin group in [DPS09, Theorem 11.7], which leads to the following question.

Question 1.8.

Can one classify the quasi-Kähler weighted right-angled Artin group?

In general, for $G$ a Kähler group, Delzant gave a complete description of $\Sigma^{1}(G;\mathbb{Z})$ in [Del10], and Suciu further reinterpreted Delzant’s results using the tropical variety of homology jump loci in [Suc21, Theorem 12.2] (see also [PS10, Theorem 16.4]). As a continuation of these results, we prove that the first inclusion in eq. 1.1 holds as equality for Kähler groups in degree $1$ . Then we derived that the BNS invariant of a Kähler group is the same as that of its maximal metabelianization, i.e.

\Sigma^{1}(G;\mathbb{Z})=\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z}),

where $G^{\prime}=[G,G]$ , and $G^{\prime\prime}=[G^{\prime},G^{\prime}]$ . For more details, see LABEL:cor:sigma_metabelian_quotient_K\"ahler. This certainly puts some restrictions on the Kähler groups. Furthermore, we summarize some properties for the Kähler group in the next proposition. Most properties listed here should be already known to the experts. For example, $(viii)\iff(ix)$ follows from Papadima and Suciu’s work [PS10, Theorem 3.6]; $(iv)\Rightarrow(vii)$ is proved by Beauville in [Bea92] (see also [Del10, Lemme 3.1] or [Bur11, Corollary 3.6]). (viii) is also related to Arapura’s work [Ara95, Property $(\mathrm{K}^{-})$ ].

Proposition 1.9.

Let $G$ be a Kähler group. Then the following are equivalent.

(i)

$\Sigma^{1}(G;\mathbb{Z})=\mathrm{S}(G)$ .
(ii)

$G^{\prime}$ is finitely generated.
(iii)

$\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z})=\mathrm{S}(G/G^{\prime\prime})$ .
(iv)

$G^{\prime}/G^{\prime\prime}$ is finitely generated.
(v)

$G/G^{\prime\prime}$ is polycyclic.
(vi)

$G/G^{\prime\prime}$ is finitely presented.
(vii)

$G/G^{\prime\prime}$ is virtually nilpotent.
(viii)

$G^{\prime}/G^{\prime\prime}\otimes_{\mathbb{Z}}\mathbbm{k}$ is of finite dimensional $\mathbbm{k}$ - for any field coefficients $\mathbbm{k}$ .
(ix)

$\mathcal{V}^{1}(G;\mathbbm{k})$ consists of only finitely many points for any algebraically closed field coefficients $\mathbbm{k}$ .

In addition to investigating Kähler groups, we apply Theorem 1.1 to the Dwyer-Fried set. In [DF87], Dwyer and Fried studied when a regular free abelian covering of a finite CW complex admits finite Betti numbers. Their findings were further developed in [PS10, Suc14, SYZ15] with field coefficients. By employing the tropical variety over $\mathbb{Z}$ , we extend some of these results to the setting of integral coefficients.

1.4 Organization

This paper is organized as follows. In section 2, we recall Bieri, Groves and Strebel’s work. In section 3, we translate their results into the language of tropical geometry. In section 4, we recall the definitions and properties of the BNSR invariants and jump ideal and give the proof of Theorem 1.1. In section 5, we compute some examples and study the Dwyer-Fried set with $\mathbb{Z}$ -coefficients. The last section 6 is devoted to applications on Kähler groups. We prove LABEL:prop_K\"ahler in LABEL:subsec_K\"ahler and Theorem 1.6 in section 6.2.

2 Bieri, Groves and Strebel’s results

In this section, we always assume that $H$ is a finitely generated abelian group with $\mathrm{rank}_{\mathbb{Z}}H=n\geqslant 1$ . Then $\mathrm{Hom}(H;\mathbb{R})\cong\mathbb{R}^{n}$ , and the character sphere

\mathrm{S}(H)=(\mathrm{Hom}(H;\mathbb{R})-\{0\})/\mathbb{R}^{+}

is topologically an $(n-1)$ -dimensional sphere. Here $\mathbb{R}^{+}$ , the set of positive real numbers, acts on $\mathrm{Hom}(H;\mathbb{R})-\{0\}$ by scalar multiplication. We will abuse the notation $\chi$ for both a nonzero character and its equivalent class $[\chi]$ in $\mathrm{S}(H)$ . For any subset $Z\subseteq\mathrm{Hom}(H;\mathbb{R})$ , denote the image of $Z-\{0\}$ in $\mathrm{S}(H)$ by $\mathrm{S}(Z)$ .

Let $R$ be a commutative Noetherian ring with unity. Then the group ring $RH$ is also commutative and Noetherian. Given any nonzero $\chi\in\mathrm{Hom}(H;\mathbb{R})$ , denote

H_{\chi}=\{h\in H\mid\chi(h)\geqslant 0\}

the associated submonoid. Then $RH_{\chi}$ is a subring of $RH$ , hence any $RH$ -module can be viewed as a $RH_{\chi}$ -module.

Following Bieri, Groves and Strebel, for a finitely generated $RH$ -module $M$ , one can attach the Sigma-invariant $\Sigma(M)\subseteq\mathrm{S}(H)$ defined as

\Sigma(M)\coloneqq\{\chi\in\mathrm{S}(H)\mid M\text{ is finitely generated over }RH_{\chi}\}

and $\Sigma^{c}(M)$ as its complementary in $\mathrm{S}(H)$ . The set $\Sigma(M)$ plays an important role in answering many algebraic questions, see [BS80, BS81, BG84] for more details.

Set

\mathscr{S}_{\chi}\coloneqq\{1+\sum_{h\in H}a_{h}\cdot h\mid\text{it is a finite sum with }a_{h}\in R\text{ and }\chi(h)>0\},

(2.1)

which is a multiplicative subset of $RH$ . Bieri and Strebel gave a complete description of $\Sigma^{c}(M)$ as follows.

Theorem 2.1 ([BS80], Proposition 2.1).

Let $R$ be a commutative Noetherian ring with unity and $H$ a finitely generated abelian group with ${\mathrm{rank}}_{\mathbb{Z}}H\geqslant 1$ . Assume that $M$ is a finitely generated $RH$ -module with its annihilator ideal denoted by $\mathrm{Ann}(M)$ . Then we have

\Sigma^{c}(M)=\{\chi\in\mathrm{S}(H)\mid\mathrm{Ann}(M)\cap\mathscr{S}_{\chi}=\emptyset\}.

Remark 2.2.

Bieri and Strebel proved the above theorem for $R=\mathbb{Z}$ , but the given proof remains valid if $\mathbb{Z}$ is generalized to $R$ , see [BS81, section 1.2]. The precise statement as in the above theorem also appeared in the proof of [BG84, Theorem 8.1].

As an application, Bieri and Strebel gave the following computational results.

Theorem 2.3 ([BS81], Theorem 1.1).

With the same notations and assumptions as in Theorem 2.1, we further assume $\sqrt{\mathrm{Ann}(M)}=\bigcap\limits_{j=1}^{q}\mathfrak{p}_{j}$ , where $\{\mathfrak{p}_{j}\}_{j=1}^{q}$ are all minimal prime ideals containing ${\mathrm{Ann}(M)}$ . Then we have

\Sigma^{c}(M)=\Sigma^{c}(RH/\mathrm{Ann}(M))=\bigcup_{j=1}^{q}\Sigma^{c}(RH/\mathfrak{p}_{j}).

In particular, $\Sigma^{c}(M)$ only depends on the radical ideal $\sqrt{\mathrm{Ann}(M)}$ .

Now the computation of $\Sigma^{c}(M)$ is reduced to the case when $M=RH/I$ with $I\subsetneq RH$ a proper ideal. Bieri, Groves and Strebel further reinterpreted $\Sigma^{c}(RH/I)$ by valuations. To explain their results, we recall the definition of valuations on rings.

Definition 2.4 ([Bou98], Chapter 4).

For a commutative ring $A$ with unity, a ring valuation $v$ on $A$ is a map $v\colon A\to\mathbb{R}_{\infty}\coloneqq\mathbb{R}\cup\{\infty\}$ such that for any $a,b\in A$ we have that

(i)

$v(ab)=v(a)+v(b)$ ,
(ii)

$v(a+b)\geqslant\min\{v(a),v(b)\}$ ,
(iii)

$v(0)=\infty$ and $v(1)=0$ .

There may be nonzero elements in $v^{-1}(\infty)$ and it is easy to see that $v^{-1}(\infty)$ is a prime ideal of $A$ . When $A$ is a field, this is the classical definition of the valuation on a field.

We summarize Bieri, Groves and Strebel’s results [BS81, Theorem 2.1] and [BG84, Theorem 8.1] as follows.

Theorem 2.5.

Let $R$ be a commutative Noetherian ring with unity and $H$ a finitely generated abelian group with ${\mathrm{rank}}_{\mathbb{Z}}H\geqslant 1$ . For a valuation $v$ on $R$ and an ideal $I\subsetneq RH$ , let $\Delta^{v}_{I}(H)$ denote the set of all real characters of $H$ induced by valuations on $RH/I$ extending $v$ , i.e.

	$\displaystyle\Delta^{v}_{I}(H)=\{\chi\in\mathrm{Hom}(H;\mathbb{R})\mid$	$\displaystyle\text{ there exists a valuation }w\colon RH/I\to\mathbb{R}_{\infty}$		(2.2)
		$\displaystyle\text{ such that }(w\circ\kappa)\|_{R}=v\text{ and }(w\circ\kappa)\|_{H}=\chi\},$

where $\kappa$ is the quotient map $RH\to RH/I$ . Then we have

\Sigma^{c}(RH/I)=\bigcup\limits_{v(R)\geqslant 0}\mathrm{S}(\Delta_{I}^{v}(H)),

(2.3)

where $v$ runs through all valuations of $R$ such that $v(R)\geqslant 0$ (we call this a non-negative valuation for short).

Remark 2.6.

Bieri and Groves described $\Sigma^{c}(M)$ without assuming that $R$ is Noetherian, see [BG84, Theorem 8.1] for more details.

Since we mainly focus later on the cases when $R$ is a field or $R=\mathbb{Z}$ , the above theorem in these two cases is explained in detail as follows.

Example 2.7.

(a)

Let $R=\mathbbm{k}$ be a field. For any $a\in\mathbbm{k}^{*}\coloneqq\mathbbm{k}-\{0\}$ , we have

$0=v(1)=v(a)+v(a^{-1}).$

If $v$ is a non-negative valuation, $v(a)\geqslant 0$ and $v(a^{-1})\geqslant 0$ . Hence the non-negative valuation $v$ can only be the trivial valuation $v_{0}$ , i.e.

$v_{0}(a)=0$ for any $a\in\mathbbm{k}^{*}$ and $v_{0}(0)=\infty$ .

Then for any ideal $I\subseteq\mathbbm{k}H$ , we have

$\Sigma^{c}(\mathbbm{k}H/I)=\mathrm{S}(\Delta^{v_{0}}_{I}(H)).$

(b)

Let $R=\mathbb{Z}$ . Then $v^{-1}(\infty)$ is a prime ideal of $\mathbb{Z}$ . All valuations on $\mathbb{Z}$ are the following:

—

If $v^{-1}(\infty)=(p)$ for $p\neq 0$ a prime integer, then for any $a\notin(p)$ and $b\in(p)$ , we have $v(a)<\infty=v(b)$ , hence $v(a+b)=\min\{v(a),v(b)\}=v(a).$ Thus the valuation $v$ factors through $\mathbb{Z}/p\mathbb{Z}$ , which is reduced to a valuation on the residue field $\mathbb{F}_{p}$ . Since for any nonzero element $x\in\mathbb{F}_{p}$ , $x^{p-1}=1$ , we have $0=v(1)=v(x^{p-1})=(p-1)v(x)$ , which means $v(x)=0$ for any $x$ nonzero. We denote this valuation on $\mathbb{Z}$ as $\hat{v}_{p}$ , and call it the mod $p$ valuation:

$\hat{v}_{p}(n)=\begin{cases}\infty,&\text{if }p\mid n\\ 0,&\text{if }p\nmid n.\end{cases}$
—

If $v^{-1}(\infty)=(0)$ , $v$ extends to a valuation on $\mathbb{Q}$ defined as $v(\frac{a}{b})=v(a)-v(b)$ . By Ostrowski’s theorem, it is equivalent to either the archimedean valuation, a $p$ -adic non-archimedean valuation $v_{p}$ or a trivial valuation $v_{0}$ . Noticing that the condition (ii) in Definition 2.4 is non-archimedean, $v$ has to be the $p$ -adic valuation $v_{p}$ or the trivial valuation $v_{0}$ .

So it is direct to see that all valuations on $\mathbb{Z}$ are nonnegative. Then for any ideal $I\subsetneq\mathbb{Z}H$ , we have

\Sigma^{c}(\mathbb{Z}H/I)=\mathrm{S}(\Delta^{v_{0}}_{I}(H))\cup\bigcup_{p\text{\ prime}}\mathrm{S}\big{(}\Delta^{v_{p}}_{I}(H)\cup\Delta^{\hat{v}_{p}}_{I}(H)\big{)},

where the set of primes $p$ in the union is finite thanks to [BG84, Theorem 8.2].

(c)

In fact, as long as $R$ is a discrete valuation domain, there are at most three types of non-negative valuations on $R$ up to multiplication by a positive real number. For more details, see [BG84, section 8.4].

The following two important theorems due to Bieri, Groves and Strebel are recorded here for later use.

Theorem 2.8 ([BG84], Corollarie 8.3 & 8.4).

Let $H$ be a finitely generated abelian group with $\mathrm{rank}_{\mathbb{Z}}H\geqslant 1$ and $R$ a Dedekind domain. For a finitely generated $RH$ -module $M$ , $\Sigma^{c}(M)$ is a finite union of rationally defined convex cones over polyhedrons on $\mathrm{S}(H)$ . In particular, $\Sigma^{c}(M)$ has dense rational points.

Theorem 2.9 ([BS80], Theorem 2.4).

Let $H$ be a finitely generated abelian group with $\mathrm{rank}_{\mathbb{Z}}H\geqslant 1$ and $M$ a finitely generated $\mathbb{Z}H$ -module. Then the abelian group underlying $M$ is finitely generated over $\mathbb{Z}$ if and only if $\Sigma^{c}(M)=\emptyset$ .

3 Connections with tropical geometry

In this section, we focus on the cases where $R$ is a field $\mathbbm{k}$ or $R=\mathbb{Z}$ . We will use terminologies in tropical geometry to re-explain Theorem 2.1 and Theorem 2.5. The readers may refer to the monograph [MS15] for the required background on tropical geometry.

3.1 Tropical variety over a valued field

Let $\mathbbm{k}$ be a fixed field endowed with a possibly trivial valuation $v:\mathbbm{k}\to\mathbb{R}_{\infty}$ . We first assume that $H$ is the free abelian group $\mathbb{Z}^{n}$ . The essential modification needed to drop the condition of freeness will be provided in Definition 3.4 later. Let $\mathbbm{k}H=\mathbbm{k}[x_{1}^{{\pm 1}},\ldots,x_{n}^{{\pm 1}}]$ be the Laurent polynomial ring.

Definition 3.1 (Tropical variety over a valued field).

For an ideal $I\subseteq\mathbbm{k}[x_{1}^{{\pm 1}},\ldots,x_{n}^{{\pm 1}}]$ , there are three ways to define the tropical variety of $I$ .

(i)

For any nonzero $f=\sum\limits_{\textbf{u}\in\mathbb{Z}^{n}}a_{\textbf{u}}x^{\textbf{u}}\in\mathbbm{k}[x_{1}^{{\pm 1}},\ldots,x_{n}^{{\pm 1}}]$ and the valuation $v$ on $\mathbbm{k}$ , the tropical polynomial ${\mathrm{trop}}_{\mathbbm{k},v}(f)\colon\mathbb{R}^{n}\to\mathbb{R}$ is defined by

{\mathrm{trop}}_{\mathbbm{k},v}(f)({\bf{w}})=\min\limits_{{\textbf{u}}\in\mathbb{Z}^{n}}\{v(a_{{\bf{u}}})+{\bf{u}}\cdot{\textbf{w}}\mid a_{\textbf{u}}\neq 0\},

(3.1)

which is a piecewise linear concave function. The tropical hypersurface associated to $f$ is defined as the set

${\mathrm{Trop}}_{\mathbbm{k},v}(f)\coloneqq\{{\textbf{w}}\in\mathbb{R}^{n}\mid\text{ the minimal in\ }\lx@cref{creftype~refnum}{eq:trop_of_polynomial}\text{ is achieved at least twice}\}$ .

The tropical variety of $I\subseteq\mathbbm{k}[x_{1}^{{\pm 1}},\ldots,x_{n}^{{\pm 1}}]$ is defined as

{\mathrm{Trop}}_{\mathbbm{k},v}(I)=\bigcap_{f\in I}{\mathrm{Trop}}_{\mathbbm{k},v}(f).

(3.2)

(ii)

Fix ${\textbf{w}}\in\mathbb{R}^{n}$ . For any nonzero $f=\sum\limits_{\textbf{u}\in\mathbb{Z}^{n}}a_{\textbf{u}}x^{\textbf{u}}\in\mathbbm{k}[x_{1}^{{\pm 1}},\ldots,x_{n}^{{\pm 1}}]$ , the initial form $\mathrm{in}_{\textbf{w},v}(f)$ is the sum of all terms in $f$ where the minimal in eq. 3.1 is achieved. The initial ideal $\mathrm{in}_{\textbf{w},v}(I)$ is the ideal generated by $\mathrm{in}_{\textbf{w},v}(f)$ where $f$ runs over $I$ . Set

{\mathrm{Trop}}_{\mathbbm{k},v}(I)\coloneqq\{\textbf{w}\in\mathbb{R}^{n}\mid\mathrm{in}_{\textbf{w},v}(I)\neq\mathbbm{k}[x_{1}^{{\pm 1}},\cdots,x_{n}^{{\pm 1}}]\}.

(iii)

Let $\overline{\mathbbm{k}}$ be an algebraically closed field extending $\mathbbm{k}$ such that the extension of $v$ to $\overline{\mathbbm{k}}$ is nontrivial, still denoted as $v$ . Such field always exists. In fact, if the valuation $v$ on $\mathbbm{k}$ is nontrivial, one can take $\overline{\mathbbm{k}}$ to be the algebraic closure of $\mathbbm{k}$ . On the other hand, if the valuation $v$ on $\mathbbm{k}$ is trivial, one can take $\overline{\mathbbm{k}}$ to be the field of Puiseux series $\bigcup_{n\geqslant 1}\mathbb{K}((t^{1/n}))$ if $\mathrm{char}(\mathbbm{k})=0$ and $\mathbb{K}((t^{\mathbb{Q}}))$ if $\mathrm{char}(\mathbbm{k})>0$ . Here $\mathbb{K}$ is an algebraic closure of $\mathbbm{k}$ . Both fields $\bigcup_{n\geqslant 1}\mathbb{K}((t^{1/n}))$ and $\mathbb{K}((t^{\mathbb{Q}}))$ have nontrivial $\mathbb{Q}$ -valued valuation, see e.g. [EKL06, Example 1.2.2]. The specific choice of $\overline{\mathbbm{k}}$ is not important, as long as it is algebraically closed with a nontrivial valuation (see [MS15, Theorem 3.2.4 and Remark 3.2.5]). The tropical variety of $I$ is then defined as the closure (under Euclidean topology) of the subset of points $(v(x_{1}),\cdots,v(x_{n}))$ where $(x_{1},\cdots,x_{n})$ belongs to the variety of the ideal $I\otimes_{\mathbbm{k}}\overline{\mathbbm{k}}$ in $(\overline{\mathbbm{k}}^{*})^{n}$ .

The following fundamental theorem of tropical algebraic geometry in [MS15, Theorem 3.2.3] shows the equivalence of the above three definitions.

Theorem 3.2.

(The Fundamental theorem of tropical algebraic geometry) Let $I$ be an ideal in $\mathbbm{k}[x_{1}^{{\pm}},\cdots,x_{n}^{{\pm}}]$ with a possible trivial valuation $v$ on $\mathbbm{k}$ . Let $Z=\mathrm{Spec}(\mathbbm{k}[x_{1}^{{\pm}},\ldots,x_{n}^{{\pm}}]/I)$ denote the corresponding subscheme. Then $Z(\overline{\mathbbm{k}})$ , the set of $\overline{\mathbbm{k}}$ -points of $Z$ , is a subvariety in $(\overline{\mathbbm{k}}^{*})^{n}$ . With the above notations and assumptions, the following three subsets of $\mathbb{R}^{n}$ coincide:

(i)

the subset ${\mathrm{Trop}}_{\mathbbm{k},v}(I)$ as defined in eq. 3.2,
(ii)

the set $\{\textbf{w}\in\mathbb{R}^{n}\mid\mathrm{in}_{\textbf{w},v}(I)\neq\mathbbm{k}[x_{1}^{{\pm}},\cdots,x_{n}^{{\pm}}]\}$ ,

(iii)

the Euclidean closure of the following set of componentwise valuations of points in $Z(\overline{\mathbbm{k}})$ :

v(Z(\overline{\mathbbm{k}}))=\{(v(x_{1}),\cdots,v(x_{n}))\in\mathbb{R}^{n}\mid(x_{1},\cdots,x_{n})\in Z(\overline{\mathbbm{k}})\}.

In particular, ${\mathrm{Trop}}_{\mathbbm{k},v}(I)$ only depends on $\sqrt{I}$ . If $\sqrt{I}=\bigcap\limits_{j=1}^{q}\mathfrak{p}_{j}$ , where $\{\mathfrak{p}_{j}\}_{j=1}^{q}$ are all minimal prime ideals containing $I$ , then

{\mathrm{Trop}}_{\mathbbm{k},v}(I)={\mathrm{Trop}}_{\mathbbm{k},v}(\sqrt{I})=\bigcup_{j=1}^{q}{\mathrm{Trop}}_{\mathbbm{k},v}(\mathfrak{p}_{j}).

We recall the structure theorem for tropical varieties in [MS15, Theorem 3.3.5] with notations and terms explained there in detail.

Theorem 3.3.

(Structure theorem for tropical variety) Let $I$ be a prime ideal in $\mathbbm{k}[x_{1}^{{\pm 1}},\cdots,x_{n}^{{\pm 1}}]$ with $\dim Z(\overline{\mathbbm{k}})=d$ . Then ${\mathrm{Trop}}_{\mathbbm{k},v}(I)$ is the support of a balanced weighted $v(\mathbbm{k}^{*})$ -valued rational polyhedral complex pure of dimension $d$ .

Next we define the tropical variety when $H$ is abelian but not necessarily free.

Definition 3.4.

(The modification from free abelian to abelian) Assume that $H$ has non-trivial torsion part and $H\cong\mathbb{Z}^{n}\oplus\mathbb{Z}/d_{1}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/d_{m}\mathbb{Z}$ . Then there exists a natural abelian group epimorphism $\psi\colon\mathbb{Z}^{n+m}\twoheadrightarrow H$ , which induces a ring epimorphism $\mathbbm{k}\mathbb{Z}^{n+m}\twoheadrightarrow\mathbbm{k}H$ and an embedding

\psi^{*}\colon\mathrm{Hom}(H;\mathbb{R})\hookrightarrow\mathrm{Hom}(\mathbb{Z}^{n+m};\mathbb{R}).

Identify $\mathbbm{k}\mathbb{Z}^{n+m}$ with $\mathbbm{k}[x_{1}^{\pm 1},\cdots,x_{n}^{\pm 1};y_{1}^{\pm 1},\cdots,y_{m}^{\pm 1}]$ . Consider $K=(y_{1}^{d_{1}}-1,\cdots,y_{m}^{d_{m}}-1)$ an ideal in $\mathbbm{k}\mathbb{Z}^{n+m}$ . Then $\mathbbm{k}H\cong\mathbbm{k}\mathbb{Z}^{n+m}/K$ is a quotient ring. For any ideal $I\subseteq\mathbbm{k}H$ , there exists a unique ideal $\tilde{I}\subseteq\mathbbm{k}\mathbb{Z}^{n+m}$ containing $K$ such that $\mathbbm{k}\mathbb{Z}^{n+m}/\tilde{I}\cong\mathbbm{k}H/I.$ For any valuation $v$ on $\mathbbm{k}\mathbb{Z}^{n+m}/\tilde{I}$ , the relation $y_{i}^{d_{i}}-1$ in the ideal $K$ gives

v(y_{i}^{d_{i}})=d_{i}\cdot v(y_{i})=v(1)=0,

which implies that $v(y_{i})=0$ for any $1\leqslant i\leqslant m$ . Since $\tilde{I}\supseteq K$ ,

{\mathrm{Trop}}_{\mathbbm{k},v}(\tilde{I})\subseteq{\mathrm{Trop}}_{\mathbbm{k},v}(K)=\psi^{*}(\mathrm{Hom}(H;\mathbb{R})).

Then we can define

{\mathrm{Trop}}_{\mathbbm{k},v}(I)\coloneqq(\psi^{*})^{-1}{\mathrm{Trop}}_{\mathbbm{k},v}(\tilde{I}).

Remark 3.5.

One can also understand ${\mathrm{Trop}}_{\mathbbm{k},v}(I)$ as follows. Consider an algebraically closed field $\overline{\mathbbm{k}}$ containing $\mathbbm{k}$ with nontrivial valuation as in Definition 3.1(iii). The variety associated with the coordinate ring $\overline{\mathbbm{k}}H$ is $\coprod(\overline{\mathbbm{k}}^{*})^{n}$ , a finite disjoint union of $(\overline{\mathbbm{k}}^{*})^{n}$ , with identity $1$ corresponding to the trivial representation $H\to\overline{\mathbbm{k}}^{*}$ . The connected component containing $1$ is an affine torus $(\overline{\mathbbm{k}}^{*})^{n}$ and any other connected component is a translation of this one, by characters of finite order induced by the torsion part of $H$ . Note that for any character of finite order, its valuation has to be $0$ . Let $Z(\overline{\mathbbm{k}})$ be the $\overline{\mathbbm{k}}$ -points of $I$ and denote its connected components as $\{Z_{j}(\overline{\mathbbm{k}})\}_{1\leqslant j\leqslant q}$ . Each connected component is contained in one of the connected components of $\coprod(\overline{\mathbbm{k}}^{*})^{n}$ . Up to translation via a torsion element, all the connected components can be considered as a subset in $(\overline{\mathbbm{k}}^{*})^{n}.$ Then the tropical variety ${\mathrm{Trop}}_{\mathbbm{k},v}(I)$ is indeed the set

\bigcup\limits_{1\leqslant j\leqslant q}\overline{\{(v(x_{1}),\cdots,v(x_{n}))\mid(x_{1},\cdots,x_{n})\in Z_{j}(\overline{\mathbbm{k}})\subset(\overline{\mathbbm{k}}^{*})^{n}\}}.

(3.3)

Einsiedler, Kapranov, and Lind first proved in [EKL06, Corollary 2.2.6] that the tropical variety is indeed the set defined by Bieri and Groves using valuations. We slightly generalize their results as follows.

Proposition 3.6.

Consider a valuation $v$ on $\mathbbm{k}$ and a finitely generated abelian group $H$ with ${\mathrm{rank}}_{\mathbb{Z}}(H)\geqslant 1$ . For an ideal $I\subsetneq\mathbbm{k}H$ , let $\Delta^{v}_{I}(H)$ denote the set considered in Theorem 2.5. Then we have

\Delta^{v}_{I}(H)={\mathrm{Trop}}_{\mathbbm{k},v}(I).

Démonstration.

If $H$ is free abelian, the claim is proved in [EKL06, Corollary 2.2.6]. Now we assume that $H$ has a non-trivial torsion part and we use the notations as in Definition 3.4. Then the claim holds if we have the following equalities:

\Delta^{v}_{I}(H)=(\psi^{*})^{-1}\Delta^{v}_{\tilde{I}}(\mathbb{Z}^{n+m})=(\psi^{*})^{-1}{\mathrm{Trop}}_{\mathbbm{k},v}(\tilde{I})={\mathrm{Trop}}_{\mathbbm{k},v}(I).

The second equality follows from the free abelian case, and the last one follows from Definition 3.4. So we are left to prove the first equality. Since $\mathbbm{k}\mathbb{Z}^{n+m}/\tilde{I}\cong\mathbbm{k}H/I,$ for any valuation $w\colon\mathbbm{k}H/I\to\mathbb{R}_{\infty}$ with $w|_{\mathbbm{k}}=v$ and $w|_{H}=\chi$ , it is also a valuation on $\mathbbm{k}\mathbb{Z}^{n+m}/\tilde{I}$ such that $w|_{\mathbbm{k}}=v$ . Since $\tilde{I}\supseteq K$ , it is easy to see that $w|_{\mathbb{Z}^{n+m}}\in\psi^{*}(\mathrm{Hom}(H;\mathbb{R}))$ . In particular, $w|_{\mathbb{Z}^{n+m}}=\psi^{*}\chi$ . This gives a one to one correspondence between $\Delta^{v}_{I}(H)$ and $(\psi^{*})^{-1}\Delta^{v}_{\tilde{I}}(\mathbb{Z}^{n+m})$ , which implies the first equality. ∎

3.2 Tropical variety over rings

In this subsection, we follow [MS15, Section 1.6] to define the tropical variety over a commutative Noetherian ring $R$ . Now $H$ is a finitely generated abelian group, not necessarily free.

Fix a character $\chi\in\mathrm{Hom}(H;\mathbb{R})$ . For any nonzero $f=\sum a_{h}h\in RH$ , we denote by $\deg_{\chi}(f)$ the minimal value of $\chi(h)$ with $a_{h}\neq 0$ , and call it the $\chi$ -degree of $f$ . The initial form $\mathrm{in}_{\chi}(f)$ is the sum of all terms $a_{h}h$ in $f$ such that $\chi(h)=\deg_{\chi}(f)$ . For an ideal $I\subseteq RH$ , its initial ideal is defined as

\mathrm{in}_{\chi}(I)\coloneqq\langle\mathrm{in}_{\chi}(f)\mid f\in I\rangle.

Following [MS15, Section 1.6], we define the tropical variety over $R$ below.

Definition 3.7 (Tropical variety over a ring).

The tropical variety over $R$ of an ideal $I\subseteq RH$ is the following set

{\mathrm{Trop}}_{R}(I)\coloneqq\{\chi\in\mathrm{Hom}(H;\mathbb{R})\mid\mathrm{in}_{\chi}(I)\neq RH\}.

Note that for the zero character $\mathrm{in}_{0}I=I$ . Hence $0\in{\mathrm{Trop}}_{R}(I)$ if and only if $I$ is a proper ideal in $RH$ , which implies that ${\mathrm{Trop}}_{R}(I)=\emptyset$ if and only if $I=RH$ . From now on we always assume that $I$ is a proper ideal. Moreover, by definition if $\chi\in{\mathrm{Trop}}_{R}(I)$ , then $r\cdot\chi\in{\mathrm{Trop}}_{R}(I)$ for any positive real number $r$ . Therefore, $\mathrm{S}({\mathrm{Trop}}_{R}(I))$ shares the same information as ${\mathrm{Trop}}_{R}(I)$ . The following result shows that the work of Bieri, Groves and Strebel can be reinterpreted by the tropical variety.

Proposition 3.8.

With the above assumptions and notations, we have

\mathrm{S}\big{(}{\mathrm{Trop}}_{R}(I)\big{)}=\Sigma^{c}(RH/I).

Moreover, if $\sqrt{I}=\bigcap\limits_{j=1}^{q}\mathfrak{p}_{j}$ , where $\{\mathfrak{p}_{j}\}_{j=1}^{q}$ are all minimal prime ideals containing $I$ , then

{\mathrm{Trop}}_{R}(I)=\bigcup_{j=1}^{q}{\mathrm{Trop}}_{R}(\mathfrak{p}_{j}).

In particular, ${\mathrm{Trop}}_{R}(I)$ only depends on the radical ideal $\sqrt{I}$ .

Démonstration.

By Theorem 2.1, $\chi\in\Sigma^{c}(RH/I)$ if and only if $\mathscr{S}_{\chi}\cap I=\emptyset$ with notation in eq. 2.1. By Definition 3.7, it is clear that $\chi\in\mathrm{S}({\mathrm{Trop}}_{R}(I))$ if and only if $\mathrm{in}_{\chi}(I)\neq RH$ . Thus we only need to show that $\mathscr{S}_{\chi}\cap I\neq\emptyset$ if and only if $\mathrm{in}_{\chi}(I)=RH$ .

One direction is clear. If $f\in\mathscr{S}_{\chi}\cap I$ , then $\mathrm{in}_{\chi}(f)=1$ and $\mathrm{in}_{\chi}(I)=RH$ . Conversely, if $\mathrm{in}_{\chi}(I)=RH$ , then there exist $f_{1},\cdots,f_{k}\in I$ and $g_{1},\cdots,g_{k}\in RH$ such that

1=\sum_{j=1}^{k}\mathrm{in}_{\chi}(f_{j})\cdot g_{j}.

Let $g^{\prime}_{j}$ denote the $\chi$ -homogeneous terms of $g_{j}$ with $\chi$ -degree being $-\deg_{\chi}(f_{j}).$ Then we have

1=\sum_{j=1}^{k}\mathrm{in}_{\chi}(f_{j})\cdot g^{\prime}_{j}

Set $f=\sum_{j=1}^{k}f_{j}\cdot g^{\prime}_{j}$ . It is clear that $f\in I$ and $\mathrm{in}_{\chi}(f)=1$ , hence $f\in\mathscr{S}_{\chi}\cap I$ . Then the first claim follows. The moreover part is a direct consequence of Theorem 2.3. ∎

When $R$ is a field $\mathbbm{k}$ , the tropical variety ${\mathrm{Trop}}_{\mathbbm{k}}(I)$ defined in Definition 3.7 is indeed ${\mathrm{Trop}}_{\mathbbm{k},v_{0}}(I)$ with the trivial valuation $v_{0}$ on $\mathbbm{k}$ defined in Definition 3.1. The claim follows from Example 2.7(a) and Proposition 3.6.

When $R=\mathbb{Z}$ , the tropical variety over $\mathbb{Z}$ can be understood using the tropical varieties over various fields with valuations in Example 2.7(b). We summarize this in the next proposition and its proof follows the idea in [BG84, section 2.1 & 2.2]

Proposition 3.9.

Let $I$ be an ideal in $\mathbb{Z}H$ . Then we have

\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(I)\big{)}=\mathrm{S}\Big{(}{\mathrm{Trop}}_{\mathbb{Q},v_{0}}(I\otimes_{\mathbb{Z}}\mathbb{Q})\cup\bigcup\limits_{\begin{subarray}{c}p\text{ prime}\\ \text{finitely many}\end{subarray}}\big{(}\mathrm{Trop}_{\mathbb{Q},v_{p}}(I\otimes_{\mathbb{Z}}\mathbb{Q})\cup\mathrm{Trop}_{\mathbb{F}_{p},\hat{v}_{p}}(I\otimes_{\mathbb{Z}}\mathbb{F}_{p})\big{)}\Big{)},

where in the union we are taking the trivial valuation $v_{0}$ on $\mathbb{Q}$ , the $p$ -adic valuation $v_{p}$ on $\mathbb{Q}$ and the trivial valuation $\hat{v}_{p}$ on $\mathbb{F}_{p}$ , respectively.

Démonstration.

By Theorem 2.5, Example 2.7(b) and Proposition 3.8, we need to show that if $v$ is one of the valuations $v_{p}$ with $p\geqslant 0$ or $\hat{v}_{p}$ with $p>0$ , then

\Delta^{v}_{I}(H)={\mathrm{Trop}}_{\mathbbm{k},v}(I\otimes_{\mathbb{Z}}\mathbbm{k})

(3.4)

for a proper field $\mathbbm{k}$ .

We first study the trivial and $p$ -adic valuations $v_{p}$ over $\mathbb{Q}$ with $p\geqslant 0$ . If $I\cap\mathbb{Z}\neq(0)$ , write $I\cap\mathbb{Z}=(m)$ with $m\neq 0$ . Then any valuation $w$ on $\mathbb{Z}H/I$ gives $w(m)=\infty$ , while $w|_{\mathbb{Z}}=v_{p}$ can not happen since $v_{p}(m)\neq\infty$ . Hence $\Delta^{v_{p}}_{I}(H)=\emptyset$ . Meanwhile, $m\in I$ implies $I\otimes_{\mathbb{Z}}\mathbb{Q}=\mathbb{Q}H$ , hence $\mathrm{Trop}_{\mathbb{Q},v_{p}}(I\otimes_{\mathbb{Z}}\mathbb{Q})=\emptyset$ . So eq. 3.4 holds under this assumption.

Now we assume that $I\cap\mathbb{Z}=(0)$ . In this case, the ring $\mathbb{Q}H/(I\otimes_{\mathbb{Z}}\mathbb{Q})$ is non-trivial. Set $S=\mathbb{Z}\backslash\{0\}$ , which is a multiplicative subset of $\mathbb{Z}$ . Since $v_{p}^{-1}(\infty)=(0)$ in $\mathbb{Z}$ , we have $S\cap(0)=\emptyset$ . Then there exists a unique valuation, still denoted as $v_{p}:S^{-1}\mathbb{Z}=\mathbb{Q}\to\mathbb{R}_{\infty}$ , given by ${v_{p}}(\frac{a}{b})=v_{p}(a)-v_{p}(b)$ , $a\in\mathbb{Z},b\in S$ , i.e. the $p$ -adic valuation on $\mathbb{Q}$ . Hence any valuation $w$ on $\mathbb{Z}H/I$ with $w|_{\mathbb{Z}}=v_{p}$ and $w|_{H}=\chi$ gives a unique valuation $w^{\prime}$ on $\mathbb{Q}H/(I\otimes_{\mathbb{Z}}\mathbb{Q})$ with $w^{\prime}|_{\mathbb{Q}}=v_{p}$ and $w^{\prime}|_{H}=\chi$ . Therefore, one obtains readily from Proposition 3.6 that

\Delta_{I}^{v_{p}}(H)=\Delta^{{v_{p}}}_{I\otimes_{\mathbb{Z}}\mathbb{Q}}(H)={\mathrm{Trop}}_{\mathbb{Q},v_{p}}(I\otimes_{\mathbb{Z}}\mathbb{Q}).

This is the first reduction step considered by Bieri and Groves in [BG84, section 2.2].

Next we study the mod $p$ -valuation $\hat{v}_{p}$ with $p>0$ . If $I\otimes_{\mathbb{Z}}\mathbb{F}_{p}=\mathbb{F}_{p}H$ , then there exists $f\in I$ such that $f=1+pf^{\prime}$ . Since $f\in I$ , any valuation $w$ on $\mathbb{Z}H/I$ gives $w(f)=\infty$ . On the other hand, $\hat{v}_{p}(p)=\infty$ implies that $w(f)=w(1+pf^{\prime})=w(1)=0$ , a contradiction. Hence $\Delta^{\hat{v}_{p}}_{I}(H)=\emptyset$ . Meanwhile, $I\otimes_{\mathbb{Z}}\mathbb{F}_{p}=\mathbb{F}_{p}H$ implies $\mathrm{Trop}_{\mathbb{F}_{p},\hat{v}_{p}}(I\otimes_{\mathbb{Z}}\mathbb{F}_{p})=\emptyset$ . So eq. 3.4 holds under this assumption.

If $I\otimes_{\mathbb{Z}}\mathbb{F}_{p}\neq\mathbb{F}_{p}H$ , then the ring $\mathbb{F}_{p}H/(I\otimes_{\mathbb{Z}}\mathbb{F}_{p})$ is non-trivial. Fix a valuation $w$ on $\mathbb{Z}H/I$ with $w|_{\mathbb{Z}}=\hat{v}_{p}$ and $w|_{H}=\chi$ . Since $\hat{v}_{p}(p)=\infty$ , $w$ can also be viewed as a valuation on $\mathbb{Z}H/\langle p,I\rangle$ , where $\langle p,I\rangle$ is the ideal generated by $p$ and $I$ . Note that $\mathbb{Z}H/\langle p,I\rangle\cong\mathbb{F}_{p}H/(I\otimes_{\mathbb{Z}}\mathbb{F}_{p})$ . Then $w$ can also be viewed as a valuation on $\mathbb{F}_{p}H/(I\otimes_{\mathbb{Z}}\mathbb{F}_{p})$ . Clearly $w|_{\mathbb{F}_{p}}$ is the trivial valuation and $w|_{H}=\chi$ . From Proposition 3.6, we have

\Delta_{I}^{\hat{v}_{p}}(H)=\Delta^{\hat{v}_{p}}_{I\otimes_{\mathbb{Z}}\mathbb{F}_{p}}(H)={\mathrm{Trop}}_{\mathbb{F}_{p},\hat{v}_{p}}(I\otimes_{\mathbb{Z}}\mathbb{F}_{p}).

This is the third reduction step considered by Bieri and Groves in [BG84, section 2.2]. ∎

Remark 3.10.

In general, we have

{\mathrm{Trop}}_{\mathbb{Z}}(I)\neq{\mathrm{Trop}}_{\mathbb{Q},v_{0}}(I\otimes_{\mathbb{Z}}\mathbb{Q})\cup\bigcup\limits_{\begin{subarray}{c}p\text{ prime}\\ \text{finitely many}\end{subarray}}\big{(}\mathrm{Trop}_{\mathbb{Q},v_{p}}(I\otimes_{\mathbb{Z}}\mathbb{Q})\cup\mathrm{Trop}_{\mathbb{F}_{p},\hat{v}_{p}}(I\otimes_{\mathbb{Z}}\mathbb{F}_{p})\big{)},

as shown by Example 3.11 and Figure 1 below. But Proposition 3.9 shows that they coincide after projection onto the unit sphere. This shows the central role of the $p$ -adic tropicalization (if it is distinct from the trivial one). Its asymptotic behavior gives the trivial tropicalization and its local behavior near the origin gives the mod $p$ tropicalization. See [BG84, Theorem C1, C2] for more details.

Example 3.11.

Consider the ideal $I=(x_{1}+x_{2}-2)\subseteq\mathbb{Z}[x_{1}^{\pm 1},x_{2}^{\pm 1}]$ . See Figure 1 below for its various tropicalizations, where Figure 1(d) is ${\mathrm{Trop}}_{\mathbb{Z}}(I)$ and Figure 1(a), Figure 1(b) and Figure 1(c) are the tropical varieties of $I$ considered in $(\mathbb{Q}[x_{1}^{\pm 1},x_{2}^{\pm 1}],v_{0})$ , $(\mathbb{F}_{2}[x_{1}^{\pm 1},x_{2}^{\pm 1}],\hat{v}_{2})$ and $(\mathbb{Q}[x_{1}^{\pm 1},x_{2}^{\pm 1}],v_{2})$ , respectively. Note that the union of Figures 1(a), 1(b) and 1(c) is different from Figure 1(d), while the projections onto the unit sphere are the same (see Figure 1(f)).

(a) The trivial Trop.

(b) The mod

2

Trop.

2

-adic Trop.

(d) The

\mathbb{Z}

-Trop.

(e) Projection of 1(a) and 1(b).

(f) Projection of 1(d) or the union of 1(a), 1(b), and 1(c).

Figure 1: Comparison of several tropicalizations.

We end this section with a property for tropical varieties. Let $\psi\colon H\twoheadrightarrow H^{\prime}$ be an epimorphism of abelian groups with kernel $N$ . It induces an embedding

\psi^{*}\colon\mathrm{Hom}(H^{\prime};\mathbb{R})\hookrightarrow\mathrm{Hom}({H};\mathbb{R})

and a ring epimorphism $\psi_{*}\colon R{H}\twoheadrightarrow RH^{\prime}$ . It is easy to see that the kernel $K$ of $\psi_{*}$ is the ideal generated by $\{n-1\mid n\in N\}$ in $R{H}$ . For an ideal $I^{\prime}\subsetneq RH^{\prime}$ , we have $R{H}/{\psi_{*}^{-1}(I^{\prime})}\cong RH^{\prime}/I^{\prime}$ .

Proposition 3.12.

With the above notations and assumptions, we have

{\mathrm{Trop}}_{R}({\psi_{*}^{-1}(I^{\prime})})=\psi^{*}({\mathrm{Trop}}_{R}(I^{\prime})).

Démonstration.

By Proposition 3.8 and Theorem 2.5, we have

${\mathrm{Trop}}_{R}(I^{\prime})=\bigcup\limits_{v(R)\geqslant 0}\Delta_{I^{\prime}}^{v}(H^{\prime})$ and ${\mathrm{Trop}}_{R}(\psi_{*}^{-1}(I^{\prime}))=\bigcup\limits_{v(R)\geqslant 0}\Delta_{\psi_{*}^{-1}(I^{\prime})}^{v}(H)$ .

Note the two facts ${\mathrm{Trop}}_{R}(K)=\psi^{*}(\mathrm{Hom}(H^{\prime};\mathbb{R}))$ and $K\subseteq\psi_{*}^{-1}(I^{\prime})$ . By the ring isomorphism $R{H}/{\psi_{*}^{-1}(I^{\prime})}\cong RH^{\prime}/I^{\prime}$ , one readily sees that ${\mathrm{Trop}}_{R}({\psi_{*}^{-1}(I^{\prime})})=\psi^{*}({\mathrm{Trop}}_{R}(I^{\prime})).$ ∎

4 Proof of Theorem 1.1

In this section, we apply the results in sections 2 and 3 to prove Theorem 1.1.

4.1 The BNSR invariants revisited

We start with a finiteness condition for chain complexes, following the approach of Farber, Geoghegan, and Schütts in [FGS10].

Definition 4.1.

Let $C=(C_{i},\partial_{i})_{i\geqslant 0}$ be a non-negatively graded chain complex over a ring $A$ . For each integer $k\geqslant 0$ , we say $C$ is of finite $k$ -type if there is a chain complex $C^{\prime}$ of finitely generated projective (left) $A$ -modules and a chain map $C^{\prime}\to C$ inducing isomorphisms $H_{i}(C^{\prime})\to H_{i}(C)$ for $i<k$ and an epimorphism $H_{k}(C^{\prime})\to H_{k}(C).$

Let $X$ be a connected finite CW-complex with the fundamental group $G$ . Denote $\widetilde{X}$ the universal covering of $X$ . The cell structure of $X$ lifts to cell structures on the universal cover $\widetilde{X}$ with $G$ -action via deck transformations. Thus, the cellular chain complex $C_{*}(\widetilde{X};\mathbb{Z})$ is a complex of finitely generated free $\mathbb{Z}G$ -modules. Given any nonzero $\chi\in\mathrm{Hom}(G;\mathbb{R})$ , the set $G_{\chi}=\{g\in G\mid\chi(g)\geqslant 0\}$ is a submonoid of $G$ , which depends only on $[\chi]\in\mathrm{S}(G)$ . Then $C_{*}(\widetilde{X};\mathbb{Z})$ can also be viewed as a complex of $\mathbb{Z}G_{\chi}$ -modules. The following definition of the BNSR invariant of $X$ can be found in [FGS10].

Definition 4.2.

For each integer $k\geqslant 0$ , the $k$ -th BNSR invariant of $X$ is given by

\Sigma^{k}(X;\mathbb{Z})\coloneqq\{\chi\in\mathrm{S}(G)\mid C_{*}(\widetilde{X};\mathbb{Z})\text{ is of finite $k$-type over }\mathbb{Z}G_{\chi}\}.

We denote by $\Sigma^{k}(X;\mathbb{Z})^{c}$ the complement of $\Sigma^{k}(X;\mathbb{Z})$ in $\mathrm{S}(G)$ . It is shown in [FGS10] that $\Sigma^{k}(X;\mathbb{Z})$ is an open subset of $\mathrm{S}(G)$ and depends only on the homotopy type of $X$ . In particular, $\Sigma^{1}(X;\mathbb{Z})$ depends only on $G$ , hence one can also denote it by $\Sigma^{1}(G;\mathbb{Z})$ . This is (almost) the BNS invariant of $G$ , which can be defined via the Cayley graph as follows. One picks a finite generating set of $G$ and let $\Gamma(G)$ be the corresponding Cayley graph of $G$ . For any $\chi\in\mathrm{S}(G)$ , let $\Gamma_{\chi}(G)$ be the full subgraph on the vertex set $G_{\chi}$ .

Definition 4.3 ([BNS87]).

Let $G$ be a finitely generated group. The BNS invariant $\Sigma^{1}(G)$ consists of $\chi\in\mathrm{S}(G)$ for which the graph $\Gamma_{\chi}(G)$ is connected.

As noted by Bieri and Renz in [BR88, Section 1.3], $\Sigma^{1}(G)=-\Sigma^{1}(G;\mathbb{Z})$ . In particular, $\Sigma^{1}(G)$ does not depend on the choice of finite generating set for $G$ . To make the notations consistent, we always use $\Sigma^{1}(G;\mathbb{Z})$ instead of $\Sigma^{1}(G)$ , and all the conclusions originally about $\Sigma^{1}(G)$ will be rewritten with respect to $\Sigma^{1}(G;\mathbb{Z})$ . The complements of the BNS invariants enjoy the following naturality property.

Proposition 4.4 (Proposition 3.3, [BNS87]).

Let $\psi\colon G\twoheadrightarrow Q$ be a surjective group homomorphism between finitely generated groups. Then the induced embedding $\psi^{*}\colon S(Q)\hookrightarrow\mathrm{S}(G)$ , restricts to an injective map $\psi^{*}\colon\Sigma^{1}(Q;\mathbb{Z})^{c}\hookrightarrow\Sigma^{1}(G;\mathbb{Z})^{c}$ .

Bieri, Neumann, and Strebel showed that the BNS invariants are important in controlling the finiteness properties of kernels of abelian quotients.

Theorem 4.5 (Theorem B1,[BNS87]).

Let $G$ be a finitely generated group and let $N$ be a normal subgroup of $G$ with an abelian quotient. Denote

{\mathrm{S}(G,N)=\{\chi\in\mathrm{S}(G)\mid\chi(N)=0\}}.

Then $N$ is finitely generated if and only if $S(G,N)\subseteq-\Sigma^{1}(G;\mathbb{Z})$ . In particular, $G^{\prime}$ is finitely generated if and only if $\mathrm{S}(G)=\Sigma^{1}(G;\mathbb{Z})$ .

4.2 Jump loci ideal and Alexander ideal

Let $H=H_{1}(X;\mathbb{Z})$ , the abelianization of $G$ . Denote $X^{H}$ the maximal abelian covering of $X$ . Let $R$ be $\mathbb{Z}$ or a field $\mathbbm{k}$ . Similarly. the cellular chain complex $X^{H}$ with $R$ -coefficients, $C_{*}(X^{H};R)$ , is a bounded complex of finitely generated free $RH$ -modules:

\cdots\to C_{i+1}(X^{H};R)\xrightarrow{\partial_{i}}C_{i}(X^{H};R)\xrightarrow{\partial_{i-1}}C_{i-1}(X^{H};R)\to\cdots\xrightarrow{\partial_{0}}C_{0}(X^{H};R)\to 0.

(4.1)

Definition 4.6.

The $i$ -th Alexander invariant $H_{i}(X^{H};R)$ is the $i$ -th homology of $C_{*}(X^{H};R)$ as chain complex of $RH$ -modules. The $i$ -th Alexander ideal $\mathrm{Ann}(H_{i}(X^{H};R))$ is the annihilator ideal of $H_{i}(X^{H};R)$ as finitely generated $RH$ -modules.

Definition 4.7.

The $i$ -th jump ideal of $X$ is defined as

\mathcal{J}^{i}(X;R)=I_{c_{i}}(\partial_{i}\oplus\partial_{i-1})

where $c_{i}=\mathrm{rank}(C_{i}(X^{H};R))$ as free $RH$ -modules and $I_{c_{i}}(-)$ denotes the ideal generated by size $c_{i}\times c_{i}$ minors of the matrix and is commonly referred to as the Fitting ideal.

When $R=\mathbbm{k}$ , the maximal spectrum of these two types of ideals are under the more well-known names: Alexander varieties and homology jump loci, see e.g. [PS10, PS14]. Moreover, Papadima and Suciu established a comparison between these two types of ideals.

Theorem 4.8 ([PS10] Theorem 3.6, or [PS14] Theorem 2.5).

With the above notations, assumptions and $R=\mathbb{Z}$ or a field $\mathbbm{k}$ , for any $k\geqslant 0$ we have

\sqrt{\mathcal{J}^{\leqslant k}(X;R)}=\sqrt{\mathrm{Ann}(H_{\leqslant k}(X^{H};R))},

where $\mathcal{J}^{\leqslant k}(X;R)=\bigcap\limits_{0\leqslant j\leqslant k}\mathcal{J}^{j}(X;R)$ and $\mathrm{Ann}(H_{\leqslant k}(X^{H};R))=\bigcap\limits_{0\leqslant j\leqslant k}\mathrm{Ann}(H_{j}(X^{H};R))$ .

Remark 4.9.

Papadima and Suciu proved the above results for $R=\mathbbm{k}$ , while their argument goes without any difficulty to $R=\mathbb{Z}$ , see the proof of [PS14, Theorem 2.5].

In homological degree one, both $\mathcal{J}^{1}(X;R)$ and $\mathrm{Ann}(H_{1}(X^{H};R))$ depend only on the fundamental group $G$ (in fact only on $G/G^{\prime\prime}$ , see e.g. [Suc14, Section 2.5]). One can thus denote $\mathcal{J}^{1}(X;R)$ by $\mathcal{J}^{1}(G;R)$ . Moreover, one may compute them by abelianizing the matrix of Fox derivatives of $G$ , see e.g. [Fox53].

If $R=\mathbbm{k}$ is an algebraically closed field, the group of $\mathbbm{k}$ -valued characters, $\mathrm{Hom}(G;\mathbbm{k}^{*})$ , is a commutative affine algebraic group. Each character $\rho$ in $\mathrm{Hom}(G;\mathbbm{k}^{*})$ defines a rank one local system on $X$ , denoted by $L_{\rho}$ . Since $\mathbbm{k}^{*}$ is abelian, we have $\mathrm{Hom}(G;\mathbbm{k}^{*})=\mathrm{Hom}(H;\mathbbm{k}^{*})$ .

Definition 4.10.

The $i$ -th homology jump loci of $X$ (over $\mathbbm{k}$ ) are defined as

\mathcal{V}^{i}(X;\mathbbm{k})\coloneqq\{\rho\in\mathrm{Hom}(G;\mathbbm{k}^{*})\mid H_{i}(X;L_{\rho})\neq 0\}.

By definition $\mathcal{V}^{i}(X;\mathbbm{k})$ is the variety of the ideal $\mathcal{J}^{i}(X;\mathbbm{k})$ . Hence $\mathcal{V}^{1}(X;\mathbbm{k})$ depends only on the fundamental group $G$ and one can denote it by $\mathcal{V}^{1}(G;\mathbbm{k})$ .

In his thesis, Sikorav reinterpreted the BNS invariant of a finitely generated group in terms of the Novikov homology in degree $1$ (see [Sik87] or [Sik17]). It was later generalized to higher degrees by Bieri and Renz in [BR88] and from groups to spaces by Farber, Geoghegan and Schütts in [FGS10]. As an application of this interpretation, Papadima and Suciu proved the following theorem, which connects the BNSR invariants with the homology jump loci of $X$ . The degree $1$ case originates in the work of Delzant [Del08, Proposition 1].

Theorem 4.11 (Theorem 10.1, [PS10]).

Let $X$ be a connected finite CW complex and let $\mathbbm{k}$ be a field. Suppose $\rho\colon G\to\mathbbm{k}^{*}$ is a multiplicative character such that $H_{i}(X;L_{\rho})\neq 0$ for some $0\leqslant i\leqslant k$ . Let $v\colon\mathbbm{k}^{*}\to\mathbb{R}$ be a valuation on $\mathbbm{k}$ such that $\chi=v\circ\rho$ is non-zero. Then we have $\chi\notin\Sigma^{k}(X;\mathbb{Z})$ .

4.3 Proof of Theorem 1.1

For a field $\mathbbm{k}$ , note that $\mathcal{J}^{\leqslant k}(X;\mathbbm{k})=\mathcal{J}^{\leqslant k}(X;\mathbb{Z})\otimes\mathbbm{k}$ . Then the second inclusion in eq. 1.1 follows directly from Definition 3.7. So we only need to prove the first inclusion in eq. 1.1.

We assume that $\mathcal{J}^{\leqslant k}(X;\mathbb{Z})$ is a proper ideal in $\mathbb{Z}H$ , otherwise the claim is vacuous. Let $\mathfrak{p}$ be a minimal prime ideal containing $\sqrt{\mathcal{J}^{\leqslant k}(X;\mathbb{Z})}$ . Fix a valuation $v$ on $\mathbb{Z}$ as studied in Example 2.7(b), which is automatically non-negative. Consider a nonzero $\chi\in\Delta^{v}_{\mathfrak{p}}(H)$ with notations as in Theorem 2.5. Then there exists a valuation $w\colon\mathbb{Z}H/\mathfrak{p}\to\mathbb{R}_{\infty}$ such that $w|_{\mathbb{Z}}=v$ and $w|_{H}=\chi$ . Note that $w^{-1}(\infty)$ is a prime ideal of $\mathbb{Z}H/\mathfrak{p}$ . Let $\mathbb{K}$ be the fractional field of the quotient domain $(\mathbb{Z}H/\mathfrak{p})/w^{-1}(\infty)$ . Then $w$ can be viewed as a valuation on $\mathbb{K}$ such that $w|_{H}=\chi$ .

Consider the following composition of maps

G\twoheadrightarrow H\hookrightarrow\mathbb{Z}H\twoheadrightarrow\mathbb{Z}H/\mathfrak{p}\to\mathbb{K},

which gives a multiplicative character $\rho\colon G\to\mathbb{K}^{*}$ . Let $L_{\rho}$ be the corresponding rank one local system on $X$ with coefficient $\mathbb{K}$ . We claim that

$H_{i}(X;L_{\rho})\neq 0$ for some $0\leqslant i\leqslant k$ .

In fact, $H_{*}(X;L_{\rho})$ can be computed by the chain complex $C_{*}(X^{H};\mathbb{Z})\otimes_{\mathbb{Z}H}\mathbb{K},$ where $\mathbb{K}$ is viewed as a $\mathbb{Z}H$ -module associated to the representation $\rho$ . Note that

\bigcap\limits_{0\leqslant j\leqslant k}\mathcal{J}^{j}(X;\mathbb{Z})=\mathcal{J}^{\leqslant k}(X;\mathbb{Z})\subseteq\sqrt{\mathcal{J}^{\leqslant k}(X;\mathbb{Z})}\subseteq\mathfrak{p}.

Then there exists some $0\leqslant i\leqslant k$ such that $\mathcal{J}^{i}(X;\mathbb{Z})\subseteq\mathfrak{p}$ . By the construction of the field $\mathbb{K}$ , we have that $\mathcal{J}^{i}(X;\mathbb{Z})\otimes_{\mathbb{Z}H}\mathbb{K}=0,$ hence $H_{i}(X;L_{\rho})\neq 0$ .

Since $\chi=w\circ\rho$ is non-zero, Theorem 4.11 shows that $\chi\notin\Sigma^{k}(X;\mathbb{Z})$ , hence

\mathrm{S}(\Delta^{v}_{\mathfrak{p}}(H))\subseteq\Sigma^{k}(X;\mathbb{Z})^{c}

for any non-negative valuation $v$ on $\mathbb{Z}$ and any minimal prime ideal $\mathfrak{p}$ containing $\sqrt{\mathcal{J}^{\leqslant k}(X;\mathbb{Z})}$ . Therefore, the first inclusion in eq. 1.1 follows from Theorems 2.5 and 3.8. The remaining properties are a direct consequence of Theorem 2.8. ∎

Remark 4.12.

Let us explain Suciu’s work [Suc21, Theorem 1.1] in our notations. Assume that $\mathbbm{k}$ is algebraically closed. Since ${\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{J}^{\leqslant k}(X;\mathbbm{k}))$ depends only on $\sqrt{\mathcal{J}^{\leqslant k}(X;\mathbbm{k})}$ and $\mathcal{V}^{\leqslant k}(X;\mathbbm{k})$ is the variety of the ideal $\mathcal{J}^{\leqslant k}(X;\mathbbm{k})$ , one can also denote ${\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{J}^{\leqslant k}(X;\mathbbm{k}))$ by ${\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{\leqslant k}(X;\mathbbm{k}))$ . When $\mathbbm{k}=\mathbb{C}$ , due to Remark 3.5 and [Suc21, Section 4], ${\mathrm{Trop}}_{\mathbb{C}}(\mathcal{V}^{\leqslant k}(X;\mathbb{C}))$ is exactly the tropical variety considered in [Suc21, Theorem 1.1]. On the other hand, if $\mathbbm{k}$ has positive characteristic, using the valued field $\mathbbm{k}((t^{\mathbb{Q}}))$ one can use Suciu’s proof to show that $\Sigma^{k}(X;\mathbb{Z})\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{\leqslant k}(X;\mathbbm{k}))\big{)}^{c}.$

4.4 Comparing with Bieri, Groves and Strebel’s results

When $k=1$ , Theorem 1.1 is essentially due to Bieri, Groves, and Strebel’s work [BS80, BS81, BG84]. We explain it in detail in this subsection.

Let $G$ be a finitely presented group. Set $G^{\prime}=[G,G]$ , and $G^{\prime\prime}=[G^{\prime},G^{\prime}]$ . Then $G/G^{\prime\prime}$ is the maximal metabelian quotient group of $G$ , and $\mathrm{S}(G)=\mathrm{S}(G/G^{\prime\prime})$ . By Proposition 4.4, we have

\Sigma^{1}(G;\mathbb{Z})\subseteq\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z}).

Consider the short exact sequence

1\to G^{\prime}/G^{\prime\prime}\to G/G^{\prime\prime}\to H\to 1,

with $H$ acting on $G^{\prime}/G^{\prime\prime}$ by conjugation. Since $G$ is finitely presented, $G^{\prime}/G^{\prime\prime}$ is finitely generated as a $\mathbb{Z}H$ -module. In fact, if $X$ is a connected CW complex such that $\pi_{1}(X)=G$ , then $G^{\prime}/G^{\prime\prime}$ is the first Alexander invariant $H_{1}(X^{H};\mathbb{Z})$ . For the metabelian group $G/G^{\prime\prime}$ , Bieri and Strebel showed in [BS80] (see also [BNS87]) that

\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z})=\Sigma(H_{1}(X^{H};\mathbb{Z})),

where $\Sigma(H_{1}(X^{H};\mathbb{Z}))$ is the Sigma invariant defined in section 2. By Proposition 3.8, we have

\Sigma(H_{1}(X^{H};\mathbb{Z}))=\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathrm{Ann}(H_{1}(X^{H};\mathbb{Z})))\big{)}^{c}.

Let $I$ be the augmentation ideal of the group ring $\mathbb{Z}H$ , i.e. $I=\langle h-1\mid h\in H\rangle$ . We have $\mathrm{S}({\mathrm{Trop}}_{\mathbb{Z}}(I))=\emptyset$ and $I=\sqrt{\mathrm{Ann}(H_{0}(X^{H};\mathbb{Z}))}$ . It follows from Theorem 4.8 that

\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathrm{Ann}(H_{1}(X^{H};\mathbb{Z})))\big{)}=\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathrm{Ann}(H_{\leqslant 1}(X^{H};\mathbb{Z})))\big{)}=\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{\leqslant 1}(X;\mathbb{Z}))\big{)}.

Putting all together, we get that

\Sigma^{1}(G;\mathbb{Z})\subseteq\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z})=\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{\leqslant 1}(X;\mathbb{Z}))\big{)}^{c},

which is exactly the first inclusion in eq. 1.1 for $k=1$ . In particular, if $G$ itself is metabelian, then the inclusion becomes an equality (for $k=1$ ).

5 Examples and application to Dwyer-Fried sets

5.1 Examples

In this subsection, we compute various examples. In the first three examples, we compare Theorem 1.1 with [Suc21, Theorem 1.1]. These three examples are all one-relator groups with two generators, whose BNS-invariants can be calculated according to Brown’s algorithm [Bro87, Section 4]. Notice here we use the invariant $\Sigma^{1}(G;\mathbb{Z})$ , which is $-\Sigma^{1}(G)$ .

Example 5.1.

Let $G=\langle a,b\mid aba^{-1}b^{-2}\rangle$ be the Baumslag–Solitar group $\mathrm{BS}(1,2)$ . This is the example considered in [Suc21, Example 8.2]. Let $\chi\colon G\to\mathbb{R}$ be the non-zero homomorphism such that $\chi(a)=1$ and $\chi(b)=0$ . Then $\mathrm{S}(G)=\{\pm\chi\}$ . The abelianization of the Fox derivative gives $x-2$ . Then one gets $\sqrt{\mathcal{J}^{\leqslant 1}(G;\mathbb{Z})}=(x-1)\cap(x-2)$ and $\mathcal{V}^{\leqslant 1}(G;\mathbb{C})=\{1,2\}$ . Hence

$\mathrm{S}({\mathrm{Trop}}_{\mathbb{Z}}(\sqrt{\mathcal{J}^{\leqslant 1}(G;\mathbb{Z})}))=\{\chi\}$ and $\mathrm{S}({\mathrm{Trop}}_{\mathbb{C}}(\mathcal{V}^{\leqslant 1}(G;\mathbb{C}))=\emptyset$ .

Calculation directly gives that $\Sigma^{1}(G;\mathbb{Z})=\{-\chi\}.$

Example 5.2.

Let $G=\langle a,b\mid a^{-1}b^{-1}ab^{2}a^{-1}b^{-1}a^{2}b^{-1}a^{-1}ba^{-1}bab^{-1}\rangle$ . This is the example in [Bro87, Page 492] and in [Suc21, Example 8.5]. The abelianization of the Fox derivative gives

\begin{pmatrix}(x^{-1}_{2}-1)(x^{-1}_{1}-1)&-x_{1}^{-1}x_{2}^{-1}(x_{1}-1)^{2}\end{pmatrix}.

Then one gets $\sqrt{\mathcal{J}^{\leqslant 1}(G;\mathbb{Z})}=(x_{1}-1)$ and $\mathcal{V}^{\leqslant 1}(G;\mathbb{C})=\{x_{1}=1\}$ . Hence

$\mathrm{S}({\mathrm{Trop}}_{\mathbb{Z}}(\sqrt{\mathcal{J}^{\leqslant 1}(G;\mathbb{Z})}))=\{(0,1),(0,-1)\}=\mathrm{S}({\mathrm{Trop}}_{\mathbb{C}}(\mathcal{V}^{\leqslant 1}(G;\mathbb{C}))$ .

Calculation via Brown’s algorithm gives that $\Sigma^{1}(G;\mathbb{Z})$ consists of two open arcs on the unit circle, joining the points $(1,0)$ to $(0,1)$ and $(0,1)$ to $(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$ . So the first inclusion in eq. 1.1 is strict in this case, as shown below in Figure 2.

(a)

\Sigma^{1}(G;\mathbb{Z})

(b)

\mathbb{Z}(\text{or\ }\mathbb{C})

-Tropical upper bound.

Figure 2: Example 5.2.

Example 5.3.

As in [PS10, Example 3.5] and [SYZ15, Lemma 10.3], given a Laurent polynomial $f(x_{1},x_{2})\in\mathbb{Z}[x_{1}^{\pm 1},x_{2}^{\pm 1}]$ , there exists a group $G$ with two generators and one relation such that its abelianization is $\mathbb{Z}^{2}$ and the abelianization of the Fox derivative is

\begin{pmatrix}f\cdot(x_{2}-1)&-f\cdot(x_{1}-1)\end{pmatrix}.

It implies that $\sqrt{\mathcal{J}^{\leqslant 1}(G,\mathbb{Z})}=(f)\cap(x_{1}-1,x_{2}-1)$ . Set $f(x_{1},x_{2})=x_{1}+x_{2}-2$ . Then Theorem 1.1 gives

\Sigma^{1}(G;\mathbb{Z})\subseteq\mathrm{S}({\mathrm{Trop}}_{\mathbb{Z}}(x_{1}+x_{2}-2))^{c}.

Suciu showed that [Suc21, Theorem 1.1]

\Sigma^{1}(G;\mathbb{Z})\subseteq\mathrm{S}({\mathrm{Trop}}_{\mathbb{C},v_{0}}(x_{1}+x_{2}-2))^{c}.

The same proof is indeed valid for any field coefficients, hence one also gets

\Sigma^{1}(G;\mathbb{Z})\subseteq\mathrm{S}({\mathrm{Trop}}_{\mathbb{F}_{2},\hat{v}_{2}}(x_{1}+x_{2}-2))^{c}.

By Example 3.11 and Figures 1(e) and 1(f), we have

\mathrm{S}({\mathrm{Trop}}_{\mathbb{F}_{2},\hat{v}_{2}}(x_{1}+x_{2}-2))\bigcup\mathrm{S}({\mathrm{Trop}}_{\mathbb{C},v_{0}}(x_{1}+x_{2}-2))\subsetneq\mathrm{S}({\mathrm{Trop}}_{\mathbb{Z}}(x_{1}+x_{2}-2)).

We end this subsection with the computations for compact Riemann orbifold groups.

Definition 5.4.

Let $C_{g}$ be a compact Riemann surface of genus $g\geqslant 1$ and let $s\geqslant 0$ be an integer. If $s>0$ , fix points $\{q_{1},\ldots,q_{s}\}$ in $C_{g}$ and assign to these points an integer weight vector ${\bm{\mu}}=(\mu_{1},\cdots,\mu_{s})$ with $\mu_{i}\geqslant 2$ . The orbifold Euler characteristic of the surface with marked points is defined as

\chi^{\mathrm{orb}}(C_{g},{\bm{\mu}})=2-2g-\sum_{j=1}^{s}(1-\dfrac{1}{\mu_{j}}).

The orbifold group $\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}})$ associated to these data has presentation as follows:

\langle x_{1},\ldots,x_{g},y_{1},\ldots,y_{g},z_{1},\ldots,z_{s}|\prod_{i=1}^{g}[x_{i},y_{i}]\cdot\prod_{j=1}^{s}z_{j}=1,z_{j}^{\mu_{j}}=1\text{ for all }1\leqslant j\leqslant s\rangle.

If $\chi^{\mathrm{orb}}(C_{g},{\bm{\mu}})=0$ , i.e. $g=1$ and $s=0$ , then $\pi_{1}(C_{g})\cong\mathbb{Z}^{2}$ . Hence $\mathcal{V}^{1}(\mathbb{Z}^{2};\mathbbm{k})=\{1\}$ for any algebraically closed field $\mathbbm{k}$ and $\Sigma^{1}(\pi_{1}(C_{g});\mathbb{Z})=S^{1}$ . So we focus on the case $\chi^{\mathrm{orb}}(C_{g},{\bm{\mu}})<0$ , i.e. either $g>1$ or $g=1$ and $s>0$ . Set $\theta({\bm{\mu}})=\frac{\prod_{j=1}^{s}\mu_{j}}{{\mathrm{lcm}}(\mu_{1},\cdots,\mu_{s})}$ . Then $H=H_{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbb{Z})$ has free abelian part $\mathbb{Z}^{2g}$ and its torsion part has order $\theta({\bm{\mu}})$ . Next we compute the BNS invariant and homology jump loci of $\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}})$ .

Proposition 5.5.

Let $\mathbbm{k}$ be an algebraically closed field with $\mathrm{char}(\mathbbm{k})=p\geqslant 0$ . With the above assumptions and notations, if $\chi^{\mathrm{orb}}(C_{g},{\bm{\mu}})<0$ , we have

\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbbm{k})=\begin{cases}{\mathrm{Hom}}(H;\mathbbm{k}^{*}),&\text{if }g>1,\text{ or }g=1\text{ and }p\\ &\text{divides some }\mu_{j},\\[5.0pt] {\mathrm{Hom}}(H;\mathbbm{k}^{*})^{\prime}\cup\{1\},&\text{if }g=1,\theta({\bm{\mu}})>1,\\ &\text{and }p\text{ does not divide any }\mu_{j},\\[5.0pt] \{1\},&\text{if }g=1,\theta({\bm{\mu}})=1,\\ &\text{and }p\text{ does not divide any }\mu_{j},\end{cases}

where ${\mathrm{Hom}}(H;\mathbbm{k}^{*})^{\prime}$ is ${\mathrm{Hom}}(H;\mathbbm{k}^{*})$ taking out the connected component containing $1$ , and we use the convention that $0$ does not divide any nonzero integer. Then

{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbbm{k}))=\begin{cases}\{0\},&\text{ if }g=1,\theta({\bm{\mu}})=1,\text{ and }p\text{ does not divide any }\mu_{j},\\ \mathbb{R}^{2g},&\text{ otherwise.}\end{cases}

Hence $\Sigma^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbb{Z})=\emptyset.$

Démonstration.

When $\mathrm{char}(\mathbbm{k})=0$ , the computations for $\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbbm{k})$ can be found in [ABCAM13, Section 2] or [Suc21, Section 10]. When $\mathrm{char}(\mathbbm{k})=p>0$ , by the Fox calculus presented in [ABCAM13, Proof of Proposition 2.11], one can compute $\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbbm{k})$ as in [LL23, Section 3.2]. Then the computations for tropical variety follow easily. Hence $\Sigma^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbb{Z})=\emptyset$ by Theorem 1.1. ∎

Remark 5.6.

In the above proof, if $\chi^{\mathrm{orb}}(C_{g},{\bm{\mu}})<0$ , we have

\Sigma^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbb{Z})=\begin{cases}\mathrm{S}({\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbbm{k})))^{c},&\text{if }g=1,\theta({\bm{\mu}})=1,\\ &\text{and }p\text{ divides some }\mu_{j}\\[5.0pt] \mathrm{S}({\mathrm{Trop}}_{\mathbb{C}}(\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbb{C})))^{c},&\text{ otherwise.}\end{cases}

This shows that the $p$ -adic tropicalization considered in Proposition 3.9 is not needed for ${\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbb{Z}))$ .

5.2 Dwyer-Fried sets

Consider an epimorphism $\nu:\pi_{1}(X)=G\twoheadrightarrow H^{\prime}$ with $H^{\prime}$ abelian. Let $X^{\nu}\to X$ be the corresponding abelian covering. Consider $\mathbbm{k}$ an algebraically closed field and denote by $\nu^{\#}\colon\mathrm{Hom}(H^{\prime};\mathbbm{k}^{*})\to\mathrm{Hom}(G;\mathbbm{k}^{*})$ the induced morphism. Suciu, Yang, and Zhao generalized Dwyer and Fried’s results (from a free abelian quotient to an abelian quotient) in [SYZ15, Theorem B] as follows:

\mathrm{dim}_{\mathbbm{k}}H_{\leqslant k}(X^{\nu};\mathbbm{k})<\infty\Longleftrightarrow\mathrm{im}(\nu^{\#})\cap\mathcal{V}^{\leqslant k}(X;\mathbbm{k})\text{\ is finite},

where $H_{\leqslant k}(X^{\nu};\mathbbm{k})=\bigoplus_{i=0}^{k}H_{i}(X^{\nu};\mathbbm{k})$ . With our notations, this conclusion can be partially generalized to the integer coefficients as follows.

Proposition 5.7.

Let $\nu:G\twoheadrightarrow H^{\prime}$ be the quotient from $G$ to an abelian group $H^{\prime}$ with ${\mathrm{rank}}_{\mathbb{Z}}H^{\prime}\geqslant 1$ . Consider $H_{i}(X^{\nu};\mathbb{Z})$ as a finitely generated $\mathbb{Z}H^{\prime}$ -module induced by the deck transformation. Then $H_{i}(X^{\nu};\mathbb{Z})$ is finitely generated over $\mathbb{Z}$ if and only if ${\mathrm{Trop}}_{\mathbb{Z}}(\mathrm{Ann}(H_{i}(X^{\nu};\mathbb{Z})))\subseteq\{0\}$ . Moreover, if ${\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{\leqslant k}(X;\mathbb{Z}))\subseteq\{0\}$ , then $H_{i}(X^{\nu};\mathbb{Z})$ is finitely generated over $\mathbb{Z}$ for any such abelian cover $\nu$ and all $0\leqslant i\leqslant k$ .

Démonstration.

The first claim follows directly from Theorem 2.9 and Proposition 3.8. For the second claim, since $H^{\prime}$ is abelian, the map $\nu$ factors through the abelianization map $G\twoheadrightarrow H$ with an epimorphism $\psi\colon H\twoheadrightarrow H^{\prime}$ . Then $\psi$ induces an embedding

\psi^{*}\colon\mathrm{Hom}(H^{\prime};\mathbb{R})\hookrightarrow\mathrm{Hom}(H;\mathbb{R})

and a ring epimorphism $\mathbb{Z}H\twoheadrightarrow\mathbb{Z}H^{\prime}$ . Similar to the discussion at the end of section 3, there exists an ideal $K\subseteq\mathbb{Z}H$ such that $\mathbb{Z}H/K\cong\mathbb{Z}H^{\prime}.$ Set $I=\sqrt{\mathcal{J}^{\leqslant k}(X;\mathbb{Z})}\subseteq\mathbb{Z}H$ and $I^{\prime}=\sqrt{\mathrm{Ann}(H_{\leqslant k}(X^{\nu};\mathbb{Z}))}\subseteq\mathbb{Z}H^{\prime}.$ Then we claim that $I^{\prime}=I\otimes_{\mathbb{Z}H}\mathbb{Z}H^{\prime}$ , which is the image of $I$ in the quotient ring $\mathbb{Z}H^{\prime}$ . In fact, one can also define the jump ideal $\mathcal{J}^{\leqslant k}(X,\nu;\mathbb{Z})$ for the cellular chain complex $C_{*}(X^{\nu};\mathbb{Z})$ as a complex of $\mathbb{Z}H^{\prime}$ -modules as in Definition 4.7. Then by Theorem 4.8 we have $\sqrt{\mathcal{J}^{\leqslant k}(X,\nu;\mathbb{Z})}=I^{\prime}$ . On the other hand, by definition we have $\mathcal{J}^{\leqslant k}(X,\nu;\mathbb{Z})=\mathcal{J}^{\leqslant k}(X;\mathbb{Z})\otimes_{\mathbb{Z}H}\mathbb{Z}H^{\prime}$ , hence $I^{\prime}=I\otimes_{\mathbb{Z}H}\mathbb{Z}H^{\prime}$ . Note that $I+K$ is the unique ideal in $\mathbb{Z}H$ such that $\mathbb{Z}H/(I+K)\cong\mathbb{Z}H^{\prime}/I^{\prime}$ . Then we have

\psi^{*}({\mathrm{Trop}}_{\mathbb{Z}}(I^{\prime}))={\mathrm{Trop}}_{\mathbb{Z}}(I+K)\subseteq{\mathrm{Trop}}_{\mathbb{Z}}(I),

where the first equality follows from Proposition 3.12. Hence ${\mathrm{Trop}}_{\mathbb{Z}}(I)\subseteq\{0\}$ implies ${\mathrm{Trop}}_{\mathbb{Z}}(I^{\prime})\subseteq\{0\}$ . Then the second claim follows from Theorem 2.9 and Proposition 3.8. ∎

Remark 5.8.

Since ${\mathrm{Trop}}_{\mathbb{Z}}(K)=\psi^{*}(\mathrm{Hom}(H^{\prime},\mathbb{R}))$ , we have the inclusion

\psi^{*}({\mathrm{Trop}}_{\mathbb{Z}}(I^{\prime}))={\mathrm{Trop}}_{\mathbb{Z}}(I+K)\subseteq{\mathrm{Trop}}_{\mathbb{Z}}(I)\cap{\mathrm{Trop}}_{\mathbb{Z}}(K),

which could be strict. For example, if $I$ is a proper ideal in $\mathbb{Z}H$ and $I^{\prime}=\mathbb{Z}H^{\prime}$ , then $0\in{\mathrm{Trop}}_{\mathbb{Z}}(I)$ and ${\mathrm{Trop}}_{\mathbb{Z}}(I^{\prime})=\emptyset$ . This happens since tropicalization does not always respect intersections even over field coefficients: if $V$ and $W$ are subvarieties of $(\mathbbm{k}^{*})^{n}$ , then ${\mathrm{Trop}}_{\mathbbm{k}}(V\cap W)\subseteq{\mathrm{Trop}}_{\mathbbm{k}}(V)\cap{\mathrm{Trop}}_{\mathbbm{k}}(W)$ , but the inclusion may be strict.

Example 5.9.

Consider the group $G$ as in Example 5.1. By the Fox calculus there, we have the following isomorphisms

G^{\prime}/G^{\prime\prime}\cong\mathbb{Z}[x^{\pm 1}]/(x-2)\cong\mathbb{Z}[\frac{1}{2}].

In particular, $\mathbb{Z}[\frac{1}{2}]$ is not finitely generated as $\mathbb{Z}$ -module, which correspondences to ${\mathrm{Trop}}_{\mathbb{Z}}(\mathrm{Ann}(G^{\prime}/G^{\prime\prime}))=\{\lambda\cdot\chi\mid\lambda\in\mathbb{R}_{\geqslant 0}\}$ . On the other hand, for any field $\mathbbm{k}$ , $G^{\prime}/G^{\prime\prime}\otimes_{\mathbb{Z}}\mathbbm{k}$ is finite dimensional, which correspondences to ${\mathrm{Trop}}_{\mathbbm{k}}(\mathrm{Ann}(G^{\prime}/G^{\prime\prime}\otimes_{\mathbb{Z}}\mathbbm{k}))=\{0\}$ .

6 Kähler groups

6.1 General results on Kähler groups

Recall that a group $G$ is called a Kähler group if it can be realized as the fundamental group of a compact Kähler manifold. Using Simpson’s Lefschetz theorem [Sim93, Theorem 1], Delzant gave a complete description of $\Sigma^{1}(G)$ for $G$ a Kähler group in [Del10, Theorem 1.1]. To explain Delzant’s results, we first recall the definition of orbifold fibrations.

Definition 6.1.

Let $X$ be a compact Kähler manifold with $\pi_{1}(X)=G$ . A holomorphic map $f\colon X\to C_{g}$ is called an orbifold fibration if $f$ is surjective with connected fibers onto a Riemann surface $C_{g}$ with genus $g\geqslant 1$ . Assume that $f$ has multiple fibers over the points $\{q_{1},\ldots,q_{s}\}$ in $C_{g}$ and let $\mu_{j}$ denote the multiplicity of the multiple fiber $f^{*}(q_{j})$ (the $\gcd$ of the coefficients of the divisor $f^{*}q_{j}$ ). Such an orbifold fibration is denoted as $f\colon X\to(C_{g},{\bm{\mu}})$ and is called hyperbolic if $\chi^{\mathrm{orb}}(C_{g},{\bm{\mu}})<0$ .

Two orbifold fibrations $f\colon X\to(C_{g},{\bm{\mu}})$ and $f^{\prime}\colon X\to(C_{g^{\prime}},{\bm{\mu}}^{\prime})$ are equivalent if there is a biholomorphic map $h\colon C_{g}\to C_{g^{\prime}}$ which sends marked points to marked points while preserving multiplicities. A compact Kähler manifold $X$ admits only finitely many equivalence classes of hyperbolic orbifold fibrations, see e.g. [Del08, Theorem 2],.

Theorem 6.2 (Theorem 1.1, [Del10]).

Let $X$ be a compact Kähler manifold with $\pi_{1}(X)=G$ . Then we have

\Sigma^{1}(G;\mathbb{Z})=\mathrm{S}\big{(}\bigcup_{f}\mathrm{im}(f^{*}\colon H^{1}(C_{g};\mathbb{R})\to H^{1}(X;\mathbb{R}))\big{)}^{c},

where the union runs over all hyperbolic orbifold fibrations of $X$ . In particular, it is a finite union. Moreover, $\Sigma^{1}(G;\mathbb{Z})=\emptyset$ if and only if there exists a hyperbolic orbifold fibration $f\colon X\to C_{g}$ such that $f^{*}\colon H^{1}(C_{g};\mathbb{R})\to H^{1}(X;\mathbb{R})$ is an isomorphism.

A hyperbolic orbifold fibration $f\colon X\to(C_{g},{\bm{\mu}})$ induces an epimorphism from $G$ to the orbifold group

f_{*}\colon G\twoheadrightarrow\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}}).

Hence it induces an embedding (see e.g. [Suc14, Lemma 2.13])

\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbbm{k})\hookrightarrow\mathcal{V}^{1}(G;\mathbbm{k}).

for any algebraically closed field coefficient $\mathbbm{k}$ . Then we have

{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(C_{g},{\bm{\mu}});\mathbbm{k}))\subseteq{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{1}(G;\mathbbm{k})).

Hence Theorem 1.1, Theorem 6.2 and Remark 5.6 give the following observation.

Proposition 6.3.

Let $G$ be a Kähler group. Then we have

\Sigma^{1}(G;\mathbb{Z})=\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G;\mathbb{Z}))\big{)}^{c}=\mathrm{S}\big{(}\bigcup_{\mathrm{char}(\mathbbm{k})=p\geqslant 0}{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{1}(G;\mathbbm{k}))\big{)}^{c},

where the last union runs over algebraically closed coefficient field $\mathbbm{k}$ with $\mathrm{char}(\mathbbm{k})=p\geqslant 0$ .

Related results of this proposition are first observed by Papadima and Suciu in [PS10, Theorem 16.4], later improved by Suciu in [Suc21, Theorem 12.2].

Denote the derived series of $G$ as $G^{\prime}=\mathcal{D}^{1}G=[G,G]$ , and $\mathcal{D}^{m+1}G=[\mathcal{D}^{m}G,\mathcal{D}^{m}G]$ for $m\geqslant 1$ , and the solvable quotient as $\mathcal{R}^{m}G=G/\mathcal{D}^{m}G$ . By LABEL:prop:abelian_bnsr_K\"ahler, we have the following observation, which may give new restrictions for the Kähler group.

Corollary 6.4.

Let $G$ be a Kähler group and $G/G^{\prime\prime}$ be its maximal metabelianization. Let $N$ be a normal subgroup of $G$ with $N\leqslant G^{\prime\prime}=\mathcal{D}^{2}G$ . Then we have

\Sigma^{1}(G;\mathbb{Z})=\Sigma^{1}(G/N;\mathbb{Z})=\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z}).

In particular, $\Sigma^{1}(G;\mathbb{Z})=\Sigma^{1}(\mathcal{R}^{m}G;\mathbb{Z})$ for any $m\geqslant 2$ .

Démonstration.

Since $\mathrm{S}(G/G^{\prime\prime})=\mathrm{S}(G)$ , the naruality property in Proposition 4.4 gives us

\Sigma^{1}(G;\mathbb{Z})\subseteq\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z}).

Since ${\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G;\mathbb{Z}))$ depends only on $\sqrt{\mathcal{J}^{1}(G;\mathbb{Z})}$ and $\sqrt{\mathcal{J}^{1}(G;\mathbb{Z})}$ depends only on $G/G^{\prime\prime}$ (see e.g. [Suc14, Section 2]), we have

{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G;\mathbb{Z}))={\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G/G^{\prime\prime};\mathbb{Z})).

On the other hand, for metabelian group $G/G^{\prime\prime}$ , by Theorem 1.1 we have

\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z})\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G/G^{\prime\prime};\mathbb{Z}))\big{)}^{c}.

Putting all together, we get that

\Sigma^{1}(G;\mathbb{Z})\subseteq\Sigma^{1}(G/G^{\prime\prime};\mathbb{Z})\subseteq\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G/G^{\prime\prime};\mathbb{Z}))\big{)}^{c}=\mathrm{S}\big{(}{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G;\mathbb{Z}))\big{)}^{c}.

Then the claim follows from LABEL:prop:abelian_bnsr_K\"ahler and Proposition 4.4. ∎

Proof of LABEL:prop_K\"ahler.

To simplify the notations, set $Q=G/G^{\prime\prime}$ and $Q^{\prime}=G^{\prime}/G^{\prime\prime}$ . We prove the proposition by four steps.

Step 1: We first prove the equivalence of $(i)-(iv)$ .

$(i)\iff(ii)$ and $(iii)\iff(iv)$ : Both follow from Theorem 4.5.

$(i)\iff(iii)$ : It follows from LABEL:cor:sigma_metabelian_quotient_K\"ahler and $\mathrm{S}(G)=\mathrm{S}(Q)$ .

Step 2: We prove $(iv)\Rightarrow(v)\Rightarrow(vi)\Rightarrow(iii)$ .

$(iv)\Rightarrow(v)$ : $Q$ is an extension of $Q^{\prime}$ by $G/G^{\prime}$ . By the assumption, both $Q^{\prime}$ and $G/G^{\prime}$ are finitely generated abelian groups, particularly polycyclic. Then by [DK18, Proposition 13.73(5)], $Q$ is polycyclic.

$(v)\Rightarrow(vi)$ : By [DK18, Proposition 13.84], every finitely generated polycyclic group is finitely presented.

$(vi)\Rightarrow(iii)$ : Note that $Q$ is a finitely generated metabelian group. By a nice theorem due to Birei and Strebel [BS80, Theorem A], we have that

$Q$ is finitely presented if and only if $\Sigma^{1}(Q;\mathbb{Z})\cup-\Sigma^{1}(Q;\mathbb{Z})=S(Q).$

Note that by LABEL:cor:sigma_metabelian_quotient_K\"ahler and Theorem 6.2, we have $\Sigma^{1}(Q;\mathbb{Z})=-\Sigma^{1}(Q;\mathbb{Z})$ . Then the claim follows.

Step 3: We prove $(iv)\Rightarrow(vii)\Rightarrow(vi)$ .

$(iv)\Rightarrow(vii)$ : If $Q^{\prime}$ is finitely generated, it follows from [Del10, Lemme 3.1] or [Bur11, Corollary 3.6] that $Q$ is virtually nilpotent.

$(vii)\Rightarrow(vi)$ : By assumption, there exists a normal subgroup $N$ of $Q$ with a short exact sequence

1\to N\overset{q}{\to}Q\to Q/N\to 1,

where $N$ is nilpotent and $Q/N$ is finite. Since $Q$ is finitely generated, so is $N$ by [DK18, Lemma 7.85]. Note that finitely generated nilpotent groups and finite groups are both finitely presented, hence $Q$ is also finitely presented by [DK18, Proposition 7.30].

Step 4: Finally we show that $(iv)\Rightarrow(viii)\Rightarrow(ix)\Rightarrow(iii)$ .

$(iv)\Rightarrow(viii)$ : It is obvious.

$(viii)\Rightarrow(ix)$ : Let $\mathbbm{k}$ an algebraically closed field. By a fact in commutative algebra (see e.g. [SYZ15, Proposition 9.3]), $Q^{\prime}\otimes_{\mathbb{Z}}\mathbbm{k}$ being finite-dimensional implies that the variety of the ideal $\mathrm{Ann}(Q^{\prime}\otimes_{\mathbb{Z}}\mathbbm{k})$ consists of at most finitely many points. Note that $Q^{\prime}\otimes_{\mathbb{Z}}\mathbbm{k}$ is the first Alexander invariant of $G$ with $\mathbbm{k}$ -coefficient. Then by Theorem 4.8, $\mathcal{V}^{1}(G;\mathbbm{k})$ consists of finitely many points.

$(ix)\Rightarrow(iii)$ : Note that Theorem 3.3 shows that taking tropicalization preserves dimension over field coefficients and ${\mathrm{Trop}}_{\mathbb{K}}(\mathcal{V}^{1}(G;\mathbbm{k}))$ is homogeneous with respect to scalar multiplication by a positive real number. If $\mathcal{V}^{1}(G;\mathbbm{k})$ consists of finitely many points, we have ${\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{1}(G;\mathbbm{k}))\subseteq\{0\}$ , hence ${\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(Q,\mathbb{Z}))\subseteq\{0\}$ due to LABEL:prop:abelian_bnsr_K\"ahler. Therefore, by the discussion in section 4.4, we have $\mathrm{S}(Q)=\Sigma^{1}(Q;\mathbb{Z})$ . ∎

6.2 Weighted right-angled Artin group

In this subsection, we classify the Kähler weighted right-angled Artin groups.

Example 6.5.

Let $\Gamma_{\ell}=(V,E,\ell)$ and $\Gamma^{\prime}_{\ell^{\prime}}=(V^{\prime},E^{\prime},\ell^{\prime})$ be two labeled graphs. Denote by $\Gamma_{\ell}\sqcup\Gamma^{\prime}_{\ell^{\prime}}$ their disjoint union. Denote by $\Gamma_{\ell}*\Gamma^{\prime}_{\ell^{\prime}}$ their join, with vertex set $V\sqcup V^{\prime}$ , edge set $E\sqcup E^{\prime}\sqcup\{\{a,a^{\prime}\}|a\in V,a^{\prime}\in V^{\prime}\}$ and weight 1 on all the joining edges $\{a,a^{\prime}\}$ . Then we have

$G_{\Gamma_{\ell}\sqcup\Gamma^{\prime}_{\ell^{\prime}}}=G_{\Gamma_{\ell}}*G_{\Gamma^{\prime}_{\ell^{\prime}}}$ and $G_{\Gamma_{\ell}*\Gamma^{\prime}_{\ell^{\prime}}}=G_{\Gamma_{\ell}}\times G_{\Gamma^{\prime}_{\ell^{\prime}}}.$

For the edge weighted graph $\Gamma_{\ell}$ , if we forget the weight on the edges, we get the finite simple graph $\Gamma$ . Let $G_{\Gamma}$ denote the corresponding right-angled Artin group. The jump loci of $G_{\Gamma}$ are completely characterized by Dimca, Papadima, and Suciu as follows (see also [PS06, Theorem 5.5, Corollary 5.6] for the corresponding results about resonance varieties).

Theorem 6.6 ([DPS09], Proposition 11.5).

Let $\Gamma$ be a finite simple graph and let $G_{\Gamma}$ denote the corresponding right-angled Artin group. Then we have

\mathcal{V}^{1}(G_{\Gamma};\mathbb{C})=\bigcup_{\begin{subarray}{c}W\subseteq V\\ \Gamma_{W}\text{ is maximally disconnected}\end{subarray}}\mathbb{T}_{W},

where the union is taken over all subsets $W\subseteq V$ such that the subgraph $\Gamma_{W}$ is maximally disconnected, i.e. $\Gamma_{W}$ is disconnected and there is no disconnected subgraph of $\Gamma$ strictly containing $\Gamma_{W}$ . Here for $W\subseteq V$ , $\mathbb{T}_{W}\subseteq\mathbb{T}_{V}=(\mathbb{C}^{*})^{|V|}$ is given by

\mathbb{T}_{W}=\{(x_{a})_{a\in V}\in(\mathbb{C}^{*})^{|V|}\mid x_{a}=1\text{ for }a\notin W\}.

In particular, $\mathcal{V}^{1}(G_{\Gamma};\mathbb{C})=(\mathbb{C}^{*})^{|V|}$ if and only if $\Gamma$ is disconnected.

By an observation, we show that $\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbb{C})$ is same as $\mathcal{V}^{1}(G_{\Gamma};\mathbb{C})$ .

Corollary 6.7.

Let $G_{\Gamma_{\ell}}$ denote the weighted right-angled Artin group associated to an edge weighted graph $\Gamma_{\ell}$ . Let $G_{\Gamma}$ denote the right-angled Artin group associated to the corresponding simple graph $\Gamma$ by forgetting the weights on edges. There is a natural isomorphism $H_{1}(G_{\Gamma_{\ell}};\mathbb{Z})\cong H_{1}(G_{\Gamma};\mathbb{Z})$ , which induces an isomorphism $\mathrm{Hom}(G_{\Gamma_{\ell}},\mathbb{C}^{*})\cong\mathrm{Hom}(G_{\Gamma},\mathbb{C}^{*})$ . Under this isomorphism, we have

\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbb{C})\cong\mathcal{V}^{1}(G_{\Gamma};\mathbb{C}).

In particular, $\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbb{C})=(\mathbb{C}^{*})^{|V|}$ if and only if $\Gamma_{\ell}$ is disconnected.

Démonstration.

The degree one jump loci can be computed by the Alexander matrix given by the Fox calculus. For an edge with weight $m$ in $G_{\Gamma_{\ell}}$ connecting vertices $a$ and $b$ , we have the relation $[a,b]^{m}=1$ in $G_{\Gamma_{\ell}}$ . The abelianization of the Fox derivative of this relation gives

\begin{pmatrix}m(1-b)&m(a-1)&0&\cdots&0\end{pmatrix}^{T},

which is a column in the Alexander matrix. Comparing with the one given by the relation $[a,b]=1$ in $G_{\Gamma}$ , they only differs by a multiplication by the nonzero integer $m$ . So the Alexander matrix for $G_{\Gamma_{\ell}}$ can be obtained from the one for $G_{\Gamma}$ by multiplying the proper (nonzero) weight for the corresponding column. Since we are computing $\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbb{C})$ and any nonzero integer is a unit in $\mathbb{C}$ , we get the same fitting ideals for $G_{\Gamma_{\ell}}$ and $G_{\Gamma}$ , hence the claim follows. ∎

Now we are ready to prove Theorem 1.6.

Proof of Theorem 1.6.

By Example 6.5, $(ii)\Rightarrow(iii)$ is easy. Note that $\langle a_{1},a_{2}|[a_{1},a_{2}]^{m}=1\rangle$ is a Kähler group, see e.g. [Ue86]. Hence so is a finite product of such groups. Then $(iii)\Rightarrow(i)$ follows. We are left to prove $(i)\Rightarrow(ii)$ by two steps. From now on, we assume that $G_{\Gamma_{\ell}}$ is a Kähler group.

Step 1: We first show that $\Gamma_{\ell}$ has to be a complete graph on an even number of vertices by mimicking the proof presented by Dimca, Papadima and Suciu in [DPS09, Theorem 11.7].

Assume that $\Gamma_{\ell}$ is not a complete graph. Then there exists $W\subseteq V$ such that the subgraph $\Gamma_{W,\ell}$ is maximally disconnected. Write $W=W_{1}\sqcup W_{2}$ with both $W_{1}$ and $W_{2}$ nonempty and no edge connecting $W_{1}$ and $W_{2}$ . Then $\Gamma_{W,\ell}=\Gamma_{W_{1},\ell}\sqcup\Gamma_{W_{2},\ell}$ and so $G_{\Gamma_{W,\ell}}=G_{\Gamma_{W_{1},\ell}}*G_{\Gamma_{W_{2},\ell}}.$ By [PS10, Proposition 13.3] we have $\mathcal{V}^{1}(G_{\Gamma_{W,\ell}};\mathbb{C})=(\mathbb{C}^{*})^{|W|}$ . Moreover, the natural group epimorphism

G_{\Gamma_{\ell}}\twoheadrightarrow G_{\Gamma_{W,\ell}}

gives two embeddings

$\mathcal{V}^{1}(G_{\Gamma_{W,\ell}};\mathbb{C})\hookrightarrow\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbb{C})$ and $H^{1}(G_{\Gamma_{W,\ell}};\mathbb{C})\hookrightarrow H^{1}(G_{\Gamma_{\ell}};\mathbb{C}).$

By Corollary 6.7, the image of the first embedding gives an irreducible component of $\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbb{C})$ . Meanwhile, the image of the second embedding is the tangent space of this irreducible component. As shown by Dimca, Papadima and Suciu in [DPS09, Theorem C], such tangent space for Kähler group is a 1-isotropic space, i.e.,

H^{1}(G_{\Gamma_{W,\ell}};\mathbb{C})\wedge H^{1}(G_{\Gamma_{W,\ell}};\mathbb{C})\to H^{2}(G_{\Gamma_{\ell}};\mathbb{C})

has a 1-dimensional image and it is a non-degenerate skew-symmetric bilinear form. Since $G_{\Gamma_{W,\ell}}=G_{\Gamma_{W_{1},\ell}}*G_{\Gamma_{W_{2},\ell}}$ , [DPS09, Lemma 9.4] gives that $H^{1}(G_{\Gamma_{W,\ell}};\mathbb{C})\wedge H^{1}(G_{\Gamma_{W,\ell}};\mathbb{C})=0$ . Hence we get a contradiction.

Step 2: Since $\Gamma_{\ell}$ is a complete graph, by Corollary 6.7 we get that $\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbb{C})=\{1\}$ . Let $X$ be a compact Kähler manifold with $\pi_{1}(X)=G_{\Gamma_{\ell}}$ . By Theorem 6.2 and Proposition 5.5, there is no orbifold fibration $f\colon X\to C_{g}$ with $g>1$ . By LABEL:prop:abelian_bnsr_K\"ahler and Theorem 6.2, we have

{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G_{\Gamma_{\ell}};\mathbb{Z}))=\bigcup_{f}\mathrm{im}f^{*}(H^{1}(C_{1};\mathbb{R})\hookrightarrow H^{1}(X;\mathbb{R})),

where the union runs over all hyperbolic orbifold fibrations $f\colon X\to C_{1}$ with $C_{1}$ a compact Riemann surface of genus 1. Hence it is a finite union of two dimensional real vector spaces.

For two orbifold fibrations $f\colon X\to C_{1}$ and $g\colon X\to C^{\prime}_{1}$ , we claim that either $\mathrm{im}f^{*}=\mathrm{im}g^{*}$ or $\mathrm{im}f^{*}\cap\mathrm{im}g^{*}=\{0\}$ . In fact, since $f$ and $g$ are both holomorphic maps between compact Kähler manifolds, both $H^{1}(C_{1};\mathbb{R})$ and $H^{1}(C^{\prime}_{1};\mathbb{R})$ carry sub-Hodge structure of $H^{1}(X;\mathbb{R})$ . Then so is $H^{1}(C_{1};\mathbb{R})\cap H^{1}(C^{\prime}_{1};\mathbb{R})$ . Hence $H^{1}(C_{1};\mathbb{R})\cap H^{1}(C^{\prime}_{1};\mathbb{R})$ has dimension either 0 or 2.

Assume that $\Gamma_{\ell}$ has two adjacent edges $\{a_{1},a_{2}\}$ and $\{a_{2},a_{3}\}$ with weights $m>1$ and $m^{\prime}>1$ , respectively. Pick a primes $p$ such that $p$ divides $m$ . Fix an algebraically closed field $\mathbbm{k}$ with $\mathrm{char}(\mathbbm{k})=p$ . Let $\Gamma_{1,2}$ denote the subgraph of $G_{\Gamma,\ell}$ with only two vertices $a_{1}$ and $a_{2}$ and the edge connecting them of weight $m$ . Note that $G_{\Gamma_{1,2}}$ is a compact orbifold group with only one marked point. Then Proposition 5.5 gives us that $\mathcal{V}^{1}(G_{\Gamma_{1,2}};\mathbbm{k})=(\mathbbm{k}^{*})^{2}$ . The natural group epimorphism $G_{\Gamma_{\ell}}\twoheadrightarrow G_{\Gamma_{1,2}}$ induces an embedding

\mathcal{V}^{1}(G_{\Gamma_{1,2}};\mathbbm{k})\hookrightarrow\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbbm{k}).

Set $\mathbb{R}_{1,2}=\{(x_{1},x_{2},0,\cdots,0)\in\mathbb{R}^{|V|}\mid x_{1},x_{2}\in\mathbb{R}\}.$ Then we have

\mathbb{R}_{1,2}\subseteq{\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{V}^{1}(G_{\Gamma_{\ell}};\mathbbm{k}))\subseteq{\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G_{\Gamma_{\ell}};\mathbb{Z})).

Since we already knew that ${\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{1}(G_{\Gamma_{\ell}};\mathbb{Z}))$ is a finite union of two-dimensional real vector spaces, $\mathbb{R}_{1,2}$ has to be one of them. Hence there exists an orbifold fibration $f\colon X\to C_{1}$ such that $\mathbb{R}_{1,2}=\mathrm{im}(f^{*}\colon H^{1}(C_{1};\mathbb{R})\hookrightarrow H^{1}(X;\mathbb{R})).$ Similarly, there exists another orbifold fibration $g\colon X\to C^{\prime}_{1}$ such that $\mathbb{R}_{2,3}=\mathrm{im}g^{*}(H^{1}(C^{\prime}_{1};\mathbb{R})\hookrightarrow H^{1}(X;\mathbb{R}))$ with $\mathbb{R}_{2,3}=\{(0,x_{2},x_{3},0,\cdots,0)\in\mathbb{R}^{|V|}\mid x_{2},x_{3}\in\mathbb{R}\}.$ But $\dim(\mathbb{R}_{1,2}\cap\mathbb{R}_{2,3})=1$ contradicts the previous claim. ∎

Remark 6.8.

For a right-angled Artin group $G_{\Gamma}$ , $\Sigma^{1}(G_{\Gamma})$ is computed in [MV95]. In particular, the first inclusion in (1.1) holds as equality for $k=1$ (see [PS06, Prposition 5.8] for a proof). It would be interesting to compute $\Sigma^{1}(G_{\Gamma_{\ell}})$ for any weighted right-angled Artin group and check if the first inclusion in (1.1) holds as equality (for $k=1$ ).

Remark 6.9.

Let $\nu\colon G_{\Gamma_{\ell}}\to\mathbb{Z}$ be the homomorphism which sends each generator $a\in V$ to 1. When $\ell(e)=1$ for all $e\in E$ , the kernel of $\nu$ is the Bestvina-Brady group. Dimca, Papadima and Suciu classified the quasi-Kähler Bestvina-Brady groups in [DPS08]. It would be interesting to know which kernel of $\nu$ for $G_{\Gamma_{\ell}}$ is quasi-Kähler.

Acknowledgments

The authors would like to thank Ziyun He, Laurentiu Maxim, Alexandru I. Suciu and Botong Wang for useful discussions, and thank Thomas Delzant for Remark 1.7. Yongqiang Liu is supported by National Key Research and Development Project SQ2020YFA070080, the Project of Stable Support for Youth Team in Basic Research Field, CAS (YSBR-001), the project “Analysis and Geometry on Bundles” of Ministry of Science and Technology of the People’s Republic of China and Fundamental Research Funds for the Central Universities. Yuan Liu is partially supported by the China Postdoctoral Science Foundation (No. 2023M744396) and the China Scholarship Council (No. 202406340174).

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