Inducing spectral gaps for the cohomological
Laplacians of $\operatorname{SL}_{n}(\mathbb{Z})$ and $\operatorname{SAut}(F_{n})$

Cohomological Laplacians were introduced by Bader and Nowak [BN20] in the context of vanishing of group cohomology with unitary coefficients. Such vanishing of the first cohomology is known to be equivalent to Kazhdan’s property (T) and an algebraic characterization for it was found by Ozawa [Oza16] for finitely generated groups. It turns out that property (T) can be described in terms of a specific degree zero Laplacian $\Delta$: Ozawa showed that a group $G$ has property (T) if and only if $\Delta^{2}-\lambda\Delta$ is a sum of squares in the real group ring $\mathbb{R}G$ for some positive spectral gap $\lambda$. Using this characterization, Kaluba, Kielak, Nowak, and Ozawa [KNO19, KKN21] were able to show property (T) for $\operatorname{SL}_{n}(\mathbb{Z})$ and $\operatorname{Aut}(F_{n})$ for $n\geq 3$ and $n\geq 5$ respectively. Estimating the spectral gap $\lambda$ from below has another advantage of prividing a lower bound for $\kappa(G,S)$, the Kazhdan constant associated to a generating set $S$ of a group $G$. It is known that $G$ has property (T) if and only if $\kappa(G,S)>0$ and this inequality does not depend on the generating set. For a symmetric generating set $S=S^{-1}$, the lower bound for Kazhdan constants is given by the formula (see [BdlHV08, Remark 5.4.7] and [KN18])

where $\lambda$ is a positive constant such that $\Delta^{2}-\lambda\Delta$ is a sum of squares.

We focus on providing lower bounds for the spectral gap of $\Delta_{1}$, the one-degree Laplacian. The existence of such a positive spectral gap for a group $G$ is equivalent to vanishing of the first cohomology of $G$ with unitary coefficients and reducibility of the second (reducibility means that the images of the differentials in the chain complex of Hilbert spaces computing the cohomology for a given unitary representation are closed), see [BN20, The Main Theorem]. Once a positive spectral gap is found for $\Delta_{1}$, it turns out to be a lower bound for the Ozawa’s spectral gap, see Remark 2.2. We acknowledge that it has been recently announced by U. Bader and R. Sauer that the reducibility condition is a consequence of vanishing of cohomology one degree lower, see [BS23].

Recently, M. Kaluba, P. W. Nowak and the author showed that for $\Delta_{1}$, the first cohomological Laplacian of $\operatorname{SL}_{3}(\mathbb{Z})$, and $\lambda=0.32$, the matrix $\Delta_{1}-\lambda I$ is a sum of squares ([KMN24, Theorem 1.1]). In this paper we show how to use a slightly stronger statement for $\operatorname{SL}_{3}(\mathbb{Z})$ to induce the spectral gap for $G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$, whenever $n\geq 3$. Our technique works as well for the case $G_{n}=\operatorname{SAut}(F_{n})$. We remark however that in such a case we were unable to obtain such a spectral gap for any particular $n$. On the other other hand, in his preprint [Nit22] M. Nitsche shows property (T) for $\operatorname{SAut}(F_{4})$. This, in combination with the work of Bader and Nowak [BN20] and the announcement of Bader and Sauer, would yield the desired gap for the Laplacian in degree one for $\operatorname{SAut}(F_{4})$. Nevertheless, this will still not allow us to induce the spectral gap since we would need a spectral gap for a specific summand of the Laplacian of $\operatorname{SAut}(F_{4})$.

Our induction technique is inspired by the idea of decomposing the generating set of $G_{n}$ into three parts: square, adjacent, and opposite, see [KKN21]. This allows us to decompose $\Delta_{1}$ for any $G_{n}$ into the sum of the appropriate parts. Denoting the adjacent part by $\operatorname{Adj}_{n}$ for any $n\geq 3$, and the generating set for $G_{n}$ by $\altmathcal{S}_{n}$, the main theorem can be stated as follows.

This article is organized as follows. In section 2 we introduce Fox derivatives which provide a setting to calculate one-degree Laplacians and define the positivity in the group ring setting in terms of sums of squares. In the next section we introduce the presentions for $G_{n}$: using the elementary matrices for $G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$ and Nielsen transvections for $G_{n}=\operatorname{SAut}(F_{n})$. Next, we decompose the Laplacian in degree one into the three parts mentioned above. In section 5 we show Theorem 1.1 using the technique of symmetrization and apply this result in the section 6 to obtain a lower bound for the spectral gap of $\Delta_{1}$ for $G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$ whenever $n\geq 3$.

We describe the model of $\Delta_{1}$ which appears in Theorem 1.1. This can be achieved by means of Fox derivatives. Further details can be found in the papers of Fox and Lyndon, [Fox53, Fox54, Lyn50, LS77].

Let $G=\langle s_{1},\ldots,s_{n}|r_{1},\ldots,r_{m}\rangle$ be a finitely presented group. The Fox derivatives $\frac{\partial}{\partial s_{j}}$ are the endomorphisms of $\mathbb{Z}F_{n}$, the group ring of the free group $F_{n}=\langle s_{1},\ldots,s_{n}\rangle$, given by the following conditions:

where $\delta_{ij}$ denotes the Kronecker delta. The Jacobian of $G$ is then the matrix $J=\left[\frac{\partial r_{i}}{\partial s_{j}}\right]\in\mathbb{M}_{m\times n}(\mathbb{Z}G)$ obtained by quotienting the Fox derivatives into the group ring $\mathbb{Z}G$. Denoting by $d$ the vertical vector $\left[1-s_{j}\right]\in\mathbb{M}_{n\times 1}(\mathbb{Z}G)$, the one-degree Laplacian $\Delta_{1}$ associated to the presentation $G=\langle s_{1},\ldots,s_{n}|r_{1},\ldots,r_{m}\rangle$ is defined by

where $\Delta_{1}^{+}=J^{*}J$ and $\Delta_{1}^{-}=dd^{*}$, and, for any matrix $M$ with entries in $\mathbb{Z}G$, the matrix $M^{*}$ is the composition of the matrix transposition with the $*$-involution on $\mathbb{Z}G$ given by $g\mapsto g^{-1}$.

By saying that a matrix $M\in\mathbb{M}_{k\times k}(\mathbb{R}G)$ has a positive spectral gap we mean that there exists a positive $\lambda$ such that $M-\lambda I_{k}$ is a sum of squares.

As noted in [KMN24, Lemma 2.1], it turns out that we have some freedom in choosing the relations which build up $\Delta_{1}^{+}$. Denote

and put $\Delta_{I}^{+}=\sum_{i\in I}J_{i}^{*}J_{i}$ for any subset of indices $I\subseteq\{1,\ldots,m\}$. Then, once $\Delta_{I}^{+}+\Delta_{1}^{-}\in\mathbb{M}_{n\times n}(\mathbb{R}G)$ has a spectral gap then $\Delta_{1}=\Delta_{I}^{+}+\Delta_{\{1,\ldots,m\}\setminus I}^{+}+\Delta_{1}^{-}\in\mathbb{M}_{n\times n}(\mathbb{R}G)$ has at least the same spectral gap.

Computing lower bounds for sepctral gaps for the first Laplacians provides the bounds for the Ozawa’s spectral gap:

In this section, we introduce the presentations of $\operatorname{SL}_{n}(\mathbb{Z})$ and $\operatorname{SAut}(F_{n})$ which we shall use to perform the Fox calculus and prove property (T) for these groups. The presentations can be found in [CRW92] and [Ger84] for $\operatorname{SL}_{n}(\mathbb{Z})$ and $\operatorname{SAut}(F_{n})$ respectively. For the latter, we provide an equivalent presentation which is easy to obtain from [Ger84].

The presentation for $\operatorname{SL}_{n}(\mathbb{Z})$ is given below.

The generator $E_{ij}$ represents the elementary matrix with $1$ at the $(i,j)$-th entry and the diagonal and zeroes elsewhere. Using Remark 2.2, we shall exclude the relation $\left(E_{12}E_{21}^{-1}E_{12}\right)^{4}$ from further considerations.

The generators of $\operatorname{SAut}(F_{n})$ are denoted by $\lambda_{ij}$ and $\rho_{ij}$ for $1\leq i\neq j\leq n$ and are called left and right Nielsen transvections respectively. The relator set contains commutators and pentagonal relations as well, along with other relations which we shall not consider, again by Remark 2.2. We only present the commutator and pentagonal relations therefore. These relations are defined for distinct indices $i,j,k,l\in\{1,\ldots,n\}$ as follows:

The transvection generators correspond in $\operatorname{SAut}(F_{n})$ to the following automorphisms of $F_{n}=\langle s_{1},\ldots,s_{n}\rangle$¹¹1We take in $\operatorname{SAut}(F_{n})$ the group operation order given by composition of automorphisms, i.e. $\alpha*\beta=\alpha\circ\beta$. Note that this is different than the convention of Gersten [Ger84]. In his paper the convention is: $\alpha*\beta=\beta\circ\alpha$. We had to reverse the order in pentagonal relators for that account.:

Let $n\geq 3$ be an integer and $G_{n}$ be $\operatorname{SL}_{n}(\mathbb{Z})$ or $\operatorname{SAut}(F_{n})$. We introduce the following decompositions of the Laplacians $\Delta_{1}^{\pm}$ of $G_{n}$:

Our decomposition is inspired by [KKN21, pp. 543 – 545].

The generators of $G_{n}$ take form $\omega_{ij}$ for $1\leq i\neq j\leq n$ and $\omega$ denoting either elementary matrices or Nielsen transvections. Then, for any $3\leq m\leq n$, there exists a natural embedding $G_{m}\hookrightarrow G_{n}$ given by $\omega_{ij}\mapsto\omega_{ij}$. From now on, when talking about the relations of $G_{n}$, we shall restrict ourselves to commutator and pentagonal relations only.

Let $C_{n}$ be the $(n-1)$-simplex with the vertices $\{1,\ldots,n\}$ and let $F(C_{n})$ denote the set of its faces, that is the subsets of the set $\{1,\ldots,n\}$. Denoting by $\altmathcal{S}_{n}$ the generator set of $G_{n}$ and by $\operatorname{Free}(\altmathcal{S}_{n})$ the free group on $\altmathcal{S}_{n}$, one can define the map

by sending an element of $\operatorname{Free}(\altmathcal{S}_{n})$ to the face on the indices of the generators occurring in that element, i.e.

In order to define $\operatorname{Sq}_{n}^{\pm}$, $\operatorname{Adj}_{n}^{\pm}$ and $\operatorname{Op}_{n}^{\pm}$ we distinguish from $F(C_{n})$ the following three subsets of faces

that is, $\operatorname{Edg}_{n}$, $\operatorname{Tri}_{n}$ and $\operatorname{Tet}_{n}$ constitute respectively the sets of edges, triangles, and tetrahedra of $C_{n}$.

The decomposition $\Delta_{1}^{-}=\operatorname{Sq}_{n}^{-}+\operatorname{Adj}_{n}^{-}+\operatorname{Op}_{n}^{-}$ is obtained as follows:

We remark that the group ring elements $d^{*}\operatorname{Sq}^{-}_{n}d$, $d^{*}\operatorname{Adj}^{-}_{n}d$, and $d^{*}\operatorname{Op}^{-}_{n}d$ are equal to the elements $\operatorname{Sq}_{n}$, $\operatorname{Adj}_{n}$, and $\operatorname{Op}_{n}$ as defined in [KKN21].

The decomposition $\Delta_{1}^{+}=\operatorname{Sq}^{+}_{n}+\operatorname{Adj}^{+}_{n}+\operatorname{Op}^{+}_{n}$ is defined, in turn, by summing over specific relators of $G_{n}$.

Note that $\left(\operatorname{Adj}^{+}_{n}\right)_{s,t}$ and $\left(\operatorname{Op}^{+}_{n}\right)_{s,t}$ can be non-zero only if $s=t$ or $\phi(s)\cup\phi(t)$ belongs to $\operatorname{Tri}_{n}$ and $\operatorname{Tet}_{n}$ respectively. We shall call $\operatorname{Sq}_{n}^{\pm}$, $\operatorname{Adj}_{n}^{\pm}$ and $\operatorname{Op}_{n}^{\pm}$ the square, adjacent and opposite part respectively.

In this section we show how to get the lower bound for spectral gap of $\operatorname{Adj}_{n}=\operatorname{Adj}_{n}^{+}+\operatorname{Adj}_{n}^{-}$ for $G_{n}$ once we assume that the lower bound for the corresponding spectral gap for $G_{m}$ is known for some $n\geq m\geq 3$. Our method is inspired by the symmetrization method used in [KKN21].