Elliptic subgroups in cyclic splittings of a free group

Given a finite set $X$, an $X$-digraph is given by the data $(V,E,\cdot_{+},\cdot_{-},\lambda)$, where:

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  $V$ and $E$ are sets;

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  $\cdot_{+},\cdot_{-}:E\rightarrow V$; and

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  $\lambda:E\rightarrow X$.

We call $V$ the vertex set and $E$ the edge set. For $e\in E$, we say that $e_{-}$ is the initial vertex of $e$ and $e_{+}$ is the terminal vertex of $e$. We call lambda the labeling function and $\lambda(e)$ the label of edge $e$.

Let $S$ be an $X$-digraph. By $VS$ and $ES$ we denote the vertex and edge sets of $S$, respectively. For $v\in VS$, we define the in-link of $v$ to be $\operatorname{lk}_{+}(v):=\{e\in ES:e_{+}=v\}$, and we say the in-hyperlink of $v$ is $\operatorname{hl}_{+}(v):=\{\lambda(e):e\in\operatorname{lk}_{+}(v)\}$. Likewise, we define the out-link of $v$ to be $\operatorname{lk}_{-}(v):=\{e\in ES:e_{-}=v\}$ and the out-hyperlink of $v$ to be $\operatorname{hl}_{-}(v):=\{\lambda(e)^{-1}:e\in\operatorname{lk}_{-}(v)\}$. The link of $v$ is $\operatorname{lk}(v):=\operatorname{lk}_{-}(v)\cup\operatorname{lk}_{+}(v)$ and the hyperlink of $v$ is $\operatorname{hl}(v):=\operatorname{hl}_{-}(v)\cup\operatorname{hl}_{+}(v)$.

We say that the degree of $v\in VS$ is $\deg_{S}(v):=\#\operatorname{lk}(v)$. If $\deg_{S}(v)=0$ we say that $v$ is isolated, and if $\deg_{S}(v)=1$ we say that $v$ is a leaf. If no vertex of $S$ is a leaf, we say that $S$ is cyclically reduced.

If $u,v\in VS$ are such that there is $e\in ES$ with $e_{-}=v$ and $e_{+}=u$, then we say that $u$ and $v$ are adjacent and that $e$ is incident to both $u$ and $v$. If $e,f\in ES$ are such that $e_{-}=f_{-}$, then we say that $e$ and $f$ are coinitial; if $e_{+}=f_{+}$, then $e$ and $f$ are coterminal. The edges $e$ and $f$ are coincident if $e$ shares an endpoint with $f$.

Let $Y\subseteq X$. A $Y$-edge is any edge with label in $Y$. The set of $Y$-edges of $S$ is denoted $E_{Y}S$.

We say that $S$ is folded at $v$ if $\lambda$ induces a bijection $\operatorname{lk}(v)\rightarrow\operatorname{hl}(v)$. The graph $S$ is folded if $S$ is folded at every vertex. If $S$ is not folded, then some pair of coterminal or coinitial edges $e,f\in ES$ share the same label. We fold these edges by identifying the pair of edge $e$ and $f$ and either the vertices $e_{-}$ and $f_{-}$ if $e$ and $f$ are coinitial or the vertices $e_{+}$ and $f_{+}$ if $e$ and $f$ are coterminal. The process of performing folds in $S$ until none remain is called folding. [TODO: put in a reference on stallings graphs and assure the reader that the end result is an invariant reduced graph.]

We may delete a leaf $v$ of $S$ by deleting $v$ and the unique edge incident to it. By repeatedly deleting leaves, we eventually arrive at a cyclically reduced $X$-digraph. We call this process cyclic reduction.

Lastly, if $S$ and $T$ are $X$-digraphs, an $X$-map is a map $S\rightarrow T$ which sends vertices to vertices, edges to edges, and preserves both orientation and label. We will assume that all of our maps of $X$-digraphs are $X$-maps. An $X$-map is an immersion if the induced map on the link of a vertex is injective for every vertex of $S$.

Given an $X$-digraph $S$, we may construct the dual of $S$, denoted $S^{*}$, by adding a set of formal inverse edges $\overline{E}S:=\{\bar{e}:e\in ES\}$ and extending $\cdot_{-},\cdot_{+},$ and $\lambda$ as follows:

By defining $\overline{(\bar{e})}=e$, the above equations are satisfied for any $e\in ES^{*}$.

A path $p$ in an $X$-digraph $S$ is a sequence of edges $p=e_{1}e_{2}\dots e_{l}$ in the dual $S^{*}$ such that $(e_{i})_{+}=(e_{i+1})_{-}$ for all $i=1,\dots,l-1$. The path $p$ is a loop if we further have that $(e_{l})_{+}=(e_{1})_{-}$. The path is immersed if $e_{i+1}\neq\bar{e}_{i}$ for all $i=1,\dots,l-1$. The label of $p$ is $\lambda(p):=\lambda(e_{1})\dots\lambda(e_{l})$. The length of $p$ is $l$. We say that an $X$-digraph $S$ is connected if there exists a path between any two vertices.

Let $H\leq_{f\!g}F(X)$. The Stallings graph representing $H$ with respect to $X$, denoted $S_{X}(H)$, is the unique $X$-digraph with basepoint such that a freely reduced word in $(X^{\pm})^{*}$ represents an element of $H$ if and only if it occurs as the label of an immersed loop of $S_{X}(H)$ beginning and ending at the basepoint.

Recall that we may construct $S_{X}(H)$ as follows. Let $h_{1},\dots,h_{k}$ be elements of $(X^{\pm})^{*}$ representing a finite set of generators of $H$. Beginning with a basepoint, denoted $1$, we construct a loop beginning and ending at $1$ with label $h_{i}$ for each $i$; let $S_{0}$ be the resulting graph. We then perform all possible folds in $S_{0}$ (in any order) to obtain a folded graph $S_{1}$. Finally, we repeatedly delete leaves of $S_{1}$ different from $1$ until no leaves remain except possibly the basepoint. The resulting graph is $S_{X}(H)$, and it is well-known that $S_{X}(H)$ is invariant with respect to the choice of generating set for $H$ as well as the order of the folds and leaf deletions.

Suppose that $S_{X}(H)$ is cyclically reduced. For any $g\in F(X)$, we may construct $S_{X}(H^{g})$ from $S_{X}(H)$ by adding a new basepoint $1^{\prime}$, a path from $1$ to $1^{\prime}$ labeled by $g$, and then folding and deleting non-basepoint leaves. Therefore whenever $S_{X}(H)$ is cyclically reduced, we will forget the basepoint and think of $S_{X}(H)$ as representing $H^{F(X)}$, the conjugacy class of $H$ in $F(X)$, rather than the single subgroup $H$.

A type I Whitehead automorphism is an automorphism $\phi\in\operatorname{Aut}F(X)$ which is induced by permutations and inversions of the set $X^{\pm}$.

A type II Whitehead automorphism is an automorphism $\phi\in\operatorname{Aut}F(X)$ for which there exists $m\in X^{\pm}$ such that $\phi(m)=m$ and

for all $x\in X$. We call $m$ the multiplier for $\phi$.

Given a type II Whitehead automorphism $\phi$ with multiplier $m$, define

Then $\phi$ is determined completely by the pair $(C,m)$, and we refer to $C$ as the cut for $\phi$.

Let $C\subseteq X^{\pm}$ be such that $m\in C$ and $m^{-1}\notin C$. We call such a $C$ an $m$-cut. For any $m\in X^{\pm}$ and $m$-cut $C$, the pair $(C,m)$ defines a type II Whitehead automorphism of $F(X)$.

More generally, if $C,D\subseteq X^{\pm}$, we say that $C$ cuts $D$ if $D$ contains an element of both $C$ and $C^{\prime}:=X^{\pm}-C$.

A hypergraph is a tuple $(V,E,\iota)$, where $V$ and $E$ are sets and $\iota:E\rightarrow\mathcal{P}(V)$, where $\mathcal{P}(V)$ denotes the power set of $V$. The elements of $V$ are called vertices and the elements of $E$ are called hyperedges. We call $\iota$ the incidence function.

Let $\Gamma$ be a hypergraph. We refer to the vertex and hyperedge sets of $\Gamma$ by $V\Gamma$ and $E\Gamma$, respectively. We will refer to the incidence function by simply $\iota$ when $\Gamma$ is clear from context. We say that a hyperedge $e\in E\Gamma$ is incident to a vertex $v\in V\Gamma$ if $v\in\iota(e)$. A pair of hyperedges $e,e^{\prime}\in E\Gamma$ are coincident if $\iota(e)\cap\iota(e^{\prime})\neq\emptyset$. Two vertices $v,v^{\prime}\in V\Gamma$ are adjacent if there is a hyperedge $e\in E\Gamma$ with $v,v^{\prime}\in\iota(e)$.

More generally, if $Y\subset V\Gamma$, we say that a hyperedge $e\in E\Gamma$ is incident to $Y$ if $\iota(e)\cap Y\neq\emptyset$. Let $Y_{1},\dots,Y_{n},Z$ be subsets of $V\Gamma$. We say that a hyperedge $e\in E\Gamma$ has type $(Y_{1},Y_{2},\dots,Y_{n};Z)$ if $e$ is incident to each $Y_{i}$ for $i=1,\dots,n$ but $e$ is not incident to $Z$. When $Z$ is empty, we will write $(Y_{1},Y_{2},\dots,Y_{n})$ instead of $(Y_{1},Y_{2},\dots,Y_{n};\emptyset)$. We denote by $[Y_{1},Y_{2},\dots,Y_{n};Z]_{\Gamma}$ the number of hyperedges of $\Gamma$ of type $(Y_{1},Y_{2},\dots,Y_{n};Z)$.

Let $Y\subseteq V\Gamma$, and let $Y^{\prime}$ denote the complement $V\Gamma-Y$. We define the capacity of $Y$ in $\Gamma$ to be the number of hyperedges of $\Gamma$ incident to both $Y$ and its complement; in the above notation,

Let $v\in V\Gamma$. The degree of $v$ in $\Gamma$ is the number of edges incident to $v$; in the above notation,

Let $S$ be a cyclically reduced $X$-digraph. We define the Whitehead hypergraph of $S$ to be the hypergraph $\Gamma(S):=(X^{\pm},VS,\operatorname{hl}:VS\rightarrow\mathcal{P}(X^{\pm}))$.

A cyclic splitting of $F(X)$ is the decomposition of $F(X)$ as the fundamental group of a graph of groups with cyclic edge groups. A free splitting of $F(X)$ is the decomposition of $F(X)$ as the fundamental group of a graph of groups with trivial edge groups. An edge map refers to a homomorphism from an edge group to a vertex subgroup in a particular graph of groups. A splitting is elementary if the corresponding graph of groups is connected and has exactly one edge. An elementary splitting is a segment splitting if the underlying graph of groups has two distinct vertices and is a loop splitting if it has only one vertex. An elementary splitting is nontrivial if it is either a loop splitting or a segment splitting in which neither edge map is an isomorphism. An elementary cyclic splitting is very small if the image of the edge group is maximal cyclic in the vertex subgroup(s).

We say that $H\leq_{f\!g}F(X)$ is elliptic in a splitting of $F(X)$ if $H$ is conjugate to a subgroup of a vertex subgroup. Subgroups which are not elliptic in a given splitting are said to be hyperbolic.

We call a subgroup $\langle A,b\rangle$ as in Proposition 2.12 a segment vertex subgroup. A subgroup $\langle U,u^{v}\rangle$ is called a loop vertex subgroup. When $A\sqcup B=X$ or $U\sqcup\{v\}=X$, we say that these vertex subgroups are standard. By $\mathcal{SV}$ and $\mathcal{LV}$ we denote the sets of standard segment and standard loop vertex subgroups, respectively.