2.1. Hyperbolically embedded pairs

In [Reference Osin19, Definition 2.9], Osin defines the notion of a collection of subgroups $\mathcal{H}$ being hyperbolically embedded into a group $G$ . This relation is denoted as $\mathcal{H} \hookrightarrow _h G$ and, in this case, we say that the pair $(G,\mathcal{H})$ is a hyperbolically embedded pair. In this article, we use the following characterization of hyperbolically embedded collection proved in [Reference Martínez-Pedroza and Rashid15] as our working definition.

Let us observe that in [Reference Martínez-Pedroza and Rashid15], Theorem 2.2 is proved for the case that $\mathcal{H}$ consists of a single infinite subgroup, and the authors observe that the argument in the case that $\mathcal{H}$ is a finite collection of infinite subgroups (such that no pair of distinct infinite subgroups in $\mathcal{H}$ are conjugate in $G$ ) follows by the same argument. Then the general case in which $\mathcal{H}$ is a finite collection of subgroups follows from the following statement: if $\mathcal{H}$ is a collection of subgroups and $K$ a finite subgroup of a group $G$ , then:

1. 1. $\mathcal{H}\hookrightarrow _h G$ if and only if $\mathcal{H}\cup \{K\} \hookrightarrow _h G$ .

2. 2. There is $(G,\mathcal{H})$ -graph if and only if there is a $(G,\mathcal{H}\cup \{K\})$ -graph.

The first statement is a direct consequence of the definition of hyperbolically embedded collection by Osin [Reference Osin19]. The if part of the second statement is trivial, and the only if part follows directly from [Reference Arora and Martínez-Pedroza2, Thm. 3.4].

2.2. Relative presentations

In [Reference Osin18, Chapter 2], Osin introduces the notions of relative presentation of a group with respect to a collection of subgroups, and relative Dehn functions. We briefly recall these notions below.

Let $G$ be a group and let $\mathcal{H}$ be a collection of subgroups. A subset $S$ of $G$ is a relative generating set of $G$ with respect to $\mathcal{H}$ if the natural homomorphism

(1) \begin{equation} F(S,\mathcal{H})=F(S)\ast_{H\in \mathcal{H}}H\longrightarrow G \end{equation}

is surjective, where $F(S)$ denotes the free group with free generating set $S$ . A relative generating set of $G$ with respect to $\mathcal{H}$ is called a generating set of the pair $(G, \mathcal{H})$ . A pair that admits a finite generating set is called a finitely generated pair. Let $R\subseteq F(S,\mathcal{H})$ be a subset that normally generates the kernel of the above homomorphism. In this case, we have a short exact sequence of groups

\begin{equation*} 1\to {\langle \!\langle R\rangle \!\rangle }\to F(S,\mathcal {H}) \to G \to 1,\end{equation*}

and the triple

(2) \begin{equation} \langle S,\mathcal{H}\ |\ R \rangle \end{equation}

is called a relative presentation of $G$ with respect to $\mathcal{H}$ , or just a presentation of the pair $(G,\mathcal{H})$ . Abusing notation, we write $G=\langle S, \mathcal{H}\mid R \rangle$ . If both $S$ and $R$ are finite we say that the pair $(G,\mathcal{H})$ is finitely presented.

Lemma 2.4. Let $G$ be a group and let $\mathcal{H}_0\sqcup \mathcal{H}$ be a collection of subgroups. Let $P$ denote the subgroup of $G$ generated by $S_0$ and the subgroups in $\mathcal{H}_0$ . If

\begin{equation*} G= \langle S_0\sqcup S, \mathcal {H}_0\cup \mathcal {H} \mid R_0\sqcup R \rangle \quad \text {and}\quad P = \langle S_0, \mathcal {H}_0 \mid R_0 \rangle \end{equation*}

then

\begin{equation*} G= \left \langle S, \mathcal {H}\cup \left \{ P \right \} \mid R' \right \rangle, \end{equation*}

where $R'$ is the image of $R$ under the natural epimorphism $\varphi \;:\; F(S_0\cup S, \mathcal{H}_0\cup \mathcal{H}) \to F(S,\mathcal{H}\cup \{ P\})$ .

Proof. Let $A=F(S, \mathcal{H})$ , $B=F(S_0, \mathcal{H}_0)$ , $K$ the normal subgroup of $B$ generated by $R_0$ and $N$ the normal subgroup of $A\ast B=F(S_0\cup S, \mathcal{H}_0\cup \mathcal{H})$ generated by $R$ . Our hypotheses imply that the natural epimorphisms $A\ast B \to G$ and $B\to P$ induce short exact sequences:

\begin{equation*} 1 \to {\langle \!\langle N,K\rangle \!\rangle }\to A\ast B \to G \to 1,\quad \text {and}\quad 1 \to K \to B \to P \to 1.\end{equation*}

Let us identify $P=B/K$ . The natural epimorphism of the statement of the lemma

\begin{equation*}\varphi \;:\; A\ast B \to A \ast (B/K)\end{equation*}

induces an isomorphism:

\begin{equation*}\hat \varphi \;:\; \frac {A\ast B}{ {\langle \!\langle N,K\rangle \!\rangle }} \to \frac {A\ast (B/K)}{ \varphi (N)} = \frac {A\ast P}{\varphi (N)}.\end{equation*}

By the definition of $N$ , we have that $\varphi (N)$ is the normal subgroup of $A\ast P$ generated by $R'=\varphi (R)$ . Therefore, the natural epimorphism $A\ast P \to G$ induces a short exact sequence:

\begin{equation*} 1 \to {\langle \!\langle R'\rangle \!\rangle }\to A\ast P \to G \to 1 \end{equation*}

which concludes the proof.

The following pair of lemmas allow us to conclude that certain amalgamated products and HNN-extensions preserve relative finite presentability.

Lemma 2.5 (Amalgamated products). For $i\in \{ 1,2\}$ , let $(G_i, \mathcal{H}_i\cup \{K_i\})$ be a pair, $\partial _i\;:\; C\to K_i$ a group monomorphism. Let $G_1\ast _C G_2$ denote the amalgamated product determined by $G_1\xleftarrow{\partial _1}C\xrightarrow{\partial _2}G_2$ , and $\mathcal{H}=\mathcal{H}_1\cup \mathcal{H}_2$ . If

\begin{equation*}G_i = \left \langle S_i, \mathcal {H}_i\cup \{K_i\}\mid R_i \right \rangle \end{equation*}

then

\begin{equation*}G_1\ast _C G_2 = \left \langle S_1\cup S_2, \mathcal {H}\cup \{ \langle K_1,K_2 \rangle \} \mid R_1\cup R_2 \right \rangle .\end{equation*}

Proof. Observe that $\langle S_1\cup S_2, \mathcal{H}\cup \{K_1, K_2\} \mid R_1\cup R_2, \partial _1(c)=\partial _2(c) \text{ for all $c\in C$} \rangle$ is a relative presentation of $G_1\ast _C G_2$ . Since the subgroup $\langle K_1, K_2 \rangle \leq G_1\ast _C G_2$ is isomorphic to the amalgamated product $K_1\ast _C K_2$ , we have that $\langle K_1,K_2 \mid \partial _1(c)=\partial _2(c) \text{ for all $c\in C$} \rangle$ is a relative presentation of $\langle K_1,K_2 \rangle$ . The proof concludes by invoking Lemma 2.4.

Lemma 2.6 (HNN-extension). Let $(G, \mathcal{H} \cup \{K,L\})$ be a pair with $K\neq L$ , $C$ a subgroup of $K$ , $\varphi \;:\; C\to L$ a group monomorphism, and let $G\ast _{\varphi }$ denote the HNN-extension $\langle G, t\mid t c t^{-1} =\varphi (c) \text{for all $c\in C$} \rangle$ . If

\begin{equation*}G=\langle S, \mathcal {H} \cup \{K,L\} \mid R \rangle \end{equation*}

then

\begin{equation*}G\ast _{\varphi }=\left \langle S,t, \mathcal {H}\cup \{ \langle K^t, L \rangle \} \mid R' \right \rangle, \end{equation*}

where $R'$ is the set of relations obtained by taking each element of $R$ and replacing all occurrences of elements $k\in K$ by words $t^{-1} k^t t$ . In particular, $R$ and $R'$ have the same cardinality.

Proof. Let $J$ denote the subgroup $K^t$ , and let $\psi \;:\; K \to J$ be the isomorphism $\psi (k)=tkt^{-1}$ . Observe that $\langle S, t, \mathcal{H}\cup \{K,L\} \mid R,\ tct^{-1}=\varphi (c) \text{ for all $c\in C$} \rangle$ is a presentation for the pair $(G\ast _\varphi, \mathcal{H}\cup \{K,L\})$ . Therefore,

\begin{equation*} G\ast _\varphi = \langle S, t, \mathcal {H}\cup \{J,L\} \mid R',\ \psi (c)=\varphi (c) \text { for all $c\in C$} \rangle .\end{equation*}

A consequence of Britton’s lemma is that the subgroup $\langle J, L \rangle \leq G\ast _\varphi$ is isomorphic to the amalgamated product $J\ast _{\varphi (C)}L$ . Hence,

\begin{equation*} \langle J, L\rangle = \langle \{J,L\} \mid \psi (c)=\varphi (c) \text { for all $c\in C$} \rangle .\end{equation*}

The proof concludes by invoking Lemma 2.4.

2.3. Relative Dehn functions

Suppose that $\langle S,\mathcal{H}\mid R\rangle$ is a finite relative presentation of the pair $(G,\mathcal{H})$ . For a word $W$ over the alphabet $\mathcal{S}=S\sqcup \bigsqcup _{H\in \mathcal{H}}(H-\{1\})$ representing the trivial element in $G$ , there is an expression:

(3) \begin{equation} W=\prod _{i=1}^k f_i^{-1}R_i f_i \end{equation}

where $R_i\in R$ and $f_i\in F(S)$ . We say a function $f\;:\; \mathbb{N}\to \mathbb{N}$ is a relative isoperimetric function of the relative presentation $\langle S,\mathcal{H}\ |\ R \rangle$ if, for any $n\in \mathbb{N}$ , and any word $W$ over the alphabet $\mathcal{S}$ of length $\leq n$ representing the trivial element in $G$ , one can write $W$ as in (3) with $k\leq f(n)$ . The smallest relative isoperimetric function of a finite relative presentation $\langle S,\mathcal{H}\ |\ R \rangle$ is called the relative Dehn function of $G$ with respect to $\mathcal{H}$ , or the Dehn function of the pair $(G, \mathcal{H})$ . This function is denoted by $\Delta _{G,\mathcal{H}}$ . Theorem 2.7 below justifies the notation $\Delta _{G,\mathcal{H}}$ for the Dehn function of a finitely presented pair $(G,\mathcal{H})$ .

For functions $f,g\;:\; \mathbb{N}\to \mathbb{N}$ , we write $f\preceq g$ if there exist constants $C,K,L\in \mathbb{N}$ such that $f(n)\leq Cg(Kn)+Ln$ for every $n$ . We say $f$ and $g$ are asymptotically equivalent, denoted as $f \asymp g$ , if $f\preceq g$ and $g\preceq f$ .

The Dehn function of a pair $(G,\mathcal{H})$ is well defined if it takes only finite values. This can be characterized in terms of fine graphs as follows.

Theorem 2.9 is essentially [Reference Hughes, Martínez-Pedroza and Saldana12, Theorem E] together with a result on Cayley–Abels graphs from [Reference Arora and Martínez-Pedroza2, Theorem H]. This is described below.

Concrete examples of Cayley–Abels graphs can be exhibited using the following construction introduced by Farb [Reference Farb10]; see also [Reference Hruska13].

That a pair $(G,\mathcal{H})$ has a well-defined function is characterized in terms of fineness of coned-off Cayley graphs.

Every coned-off Cayley graph $\hat \Gamma (G,\mathcal{H},S)$ with $S$ a finite relative generating set is a Cayley–Abels graph. The following result implies that coned-off Cayley graphs are, up to quasi-isometry, independent of the choice of finite generating set, and we denote them by $\hat \Gamma (G, \mathcal{H})$ . Observe now that Theorem 2.9 also follows from the following result.

4.1. Pushouts in the category of $G$ -sets

Let $\phi \;:\; R\to S$ and $\psi \;:\; R \to T$ be $G$ -maps. The pushout of $\phi$ and $\psi$ is defined as follows. Let $Z$ be the $G$ -set obtained as the quotient of the disjoint union of $G$ -sets $S\sqcup T$ by the equivalence relation generated by all pairs $s\sim t$ with $s\in S$ and $t\in T$ satisfying that there is $r\in R$ such that $\phi (r)=s$ and $\psi (r)=t$ . There are canonical $G$ -maps $\imath \;:\; S \to Z$ and $\jmath \;:\; T \to Z$ such that $\imath \circ \phi = \jmath \circ \psi$ . This construction satisfies the universal property of pushouts in the category of $G$ -sets.

Proposition 4.1. Let $\phi \;:\; R\to S$ and $\psi \;:\; R \to T$ be $G$ -maps. Consider the pushout

of $\phi$ and $\psi$ . Suppose there is $r\in R$ such that $R=G.r$ . If $s=\phi (r)$ , $t=\psi (r)$ and $z=\imath (s)$ , then the $G$ -stabilizer $G_z$ equals the subgroup $\langle G_s,G_t \rangle$ .

Proof. Since $\imath$ and $\jmath$ are $G$ -maps, $\langle G_s,G_t \rangle \leq G_z$ . Conversely, let $g\in G_z$ . If $g\in G_s$ then $g\in \langle G_s, G_t\rangle$ . Suppose $g\not \in G_s$ .

Let $r_0$ denote the element $r\in R$ in the statement, in particular, $s=\phi (r_0)$ , $t=\psi (r_0)$ and $R=G.r_0$ . Since $\jmath (t)=\imath (g.s)$ , the definition of $Z$ as a collection of equivalence classes in $S\sqcup T$ implies that there is a sequence $r'_{\!\!0},r_1,r'_{\!\!1}\ldots, r_k, r'_{\!\!k}$ of elements of $R$ such that

\begin{equation*} t=\psi (r'_{\!\!0}),\ \phi (r'_{\!\!0})=\phi (r_1),\ \psi (r_1)=\psi (r'_{\!\!1}),\ \ldots,\ \psi (r_k)=\psi (r'_{\!\!k}),\ \phi (r'_{\!\!k})=g.s.\end{equation*}

Let $s_i=\phi (r'_{\!\!i-1})=\phi (r_i)$ and $t_i=\psi (r_i)=\psi (r'_{\!\!i})$ . Since $R=G.r_0$ , there are elements $a_0,a_1,\ldots,a_k$ and $b_0,b_1,\ldots, b_{k-1}$ of $G$ such that

\begin{equation*} a_i.r_i=r'_{\!\!i} \quad \text {and} \quad b_j.r'_{\!\!j}=r_{j+1} \end{equation*}

for $0\leq i\leq k$ and $0\leq j\lt k$ . Then

\begin{equation*}g.s=\phi (r'_{\!\!k})=\phi (a_k b_{k-1} a_{k-1} \ldots b_0a_0.r_0)= a_k b_{k-1} a_{k-1} \ldots b_0a_0.s \end{equation*}

and hence $ a_k b_{k-1} a_{k-1} \ldots b_0a_0 \in gG_s.$ Since $G_s \leq \langle G_s, G_t \rangle$ , to prove that $g\in \langle G_s, G_t \rangle$ is enough to show that $a_i, b_j \in \langle G_s, G_t \rangle$ . We will argue by induction.

First note that since $\phi$ and $\psi$ are $G$ -maps

\begin{equation*} a_i.s_i = s_{i+1} \quad \text {and} \quad b_j.t_{j}=t_{j+1 },\end{equation*}

and hence

\begin{equation*} G_{s_{i+1}}= a_iG_{s_i}a_i^{-1} \quad \text {and} \quad G_{t_{j+1}}=b_jG_{t_{j}}b_j^{-1}.\end{equation*}

Moreover, $t_i=\psi (r_i)=\psi (r'_{\!\!i})=\psi (a_i.r_i)=a_i.t_i$ implies

\begin{equation*}a_i\in G_{t_i},\end{equation*}

and analogously $s_{j+1}=\phi (r_{j+1})=\phi (b_j.r'_{\!\!j})=b_j.s_{j+1}$ implies

\begin{equation*}b_j\in G_{s_{j+1}}.\end{equation*}

Since $t_0=t$ and $s_0=s$ , we have that

\begin{equation*}a_0\in G_{t_0} \leq \langle G_s, G_t \rangle, \quad \text {and} \quad b_0\in G_{s_1}=a_0G_{s_0}a_0^{-1} \leq \langle G_s, G_t \rangle .\end{equation*}

Suppose $i\lt k$ , $a_i,b_i\in \langle G_s, G_t \rangle$ , $G_{s_i}\leq \langle G_s, G_t \rangle$ and $G_{t_i}\leq \langle G_s, G_t \rangle$ . Then

\begin{equation*} a_{i+1} \in G_{t_{i+1}} = b_iG_{t_i}b_i^{-1} \leq \langle G_s, G_t \rangle,\end{equation*}

and hence

\begin{equation*} G_{s_{i+1}}= a_iG_{s_{i}}a_i^{-1} \leq \langle G_s, G_t \rangle .\end{equation*}

In the case that $i+1\lt k$ ,

\begin{equation*} b_{i+1} \in G_{s_{i+2}} = a_{i+1}G_{s_{i+1}}a_{i+1}^{-1} \leq \langle G_s, G_t \rangle. \end{equation*}

Therefore, by induction, $a_i,b_j\in \langle G_s, G_t \rangle$ for $0\leq i\leq k$ and $0\leq j\lt k$ .

4.2. Extending actions on sets

In the case that $K$ is a subgroup of $G$ and $S$ is a $K$ -set, one can extend the $K$ -action on $S$ to a $G$ -set $G\times _K S$ that we now describe. Up to isomorphism of $K$ -sets, we can assume that $S$ is a disjoint union of $K$ -sets:

\begin{equation*} S = \bigsqcup _{i\in I} K/K_i \end{equation*}

where $K/K_i$ is the $K$ -set consisting of left cosets of a subgroup $K_i$ of $K$ . Then the $G$ -set $G\times _K S$ is defined as a disjoint union of $G$ -sets:

\begin{equation*} G\times _K S \;:\!=\; \bigsqcup _{i\in I} G/K_i .\end{equation*}

Observe that the canonical $K$ -map

\begin{equation*} \imath \;:\; S \to G\times _K S, \qquad K_i \mapsto K_i\end{equation*}

is injective. This construction satisfies a number of useful properties that we summarize in the following proposition.

For $n$ a natural number and a set $X$ , let $[X]^n$ denote the collection of subsets of $X$ of cardinality $n$ . If $X$ is a $G$ -set, then $[X]^n$ is a $G$ -set with action defined as $g.\{x_1,\ldots,x_n\}=\{g.x_1,\ldots,g.x_n\}$ .

Proof. The first four statements are observations. For the fifth statement, suppose $\tilde f(\imath (s_1) ) = \tilde f(g.\imath (s_2))$ . Then $f( s_1 )=g.f( s_2 )$ . Since the map $S/K\to T/G$ induced by $f$ is injective, we have that $s_1$ and $s_2$ are in the same $K$ -orbit in $S$ , say $s_2=k.s_1$ for $k\in K$ . It follows that $f( s_1 )=gk.f( s_1 )$ , and since $K_{s_1} = G_{f(s_1)}$ , we have that $gk\in K_{s_1}$ . Therefore $\imath (s_1) =\imath (gk.s_1)= g.\imath (ks_1) =g.\imath (s_2)$ .

The sixth statement is proved as follows. The $K$ -map $\imath \;:\; S \to G\times _K S$ naturally induces a $K$ -map $\bar \imath \;:\; [S]^n \to [G\times _K S]^n$ . By the third statement, there is a unique $G$ -map $\hat \imath \;:\; G\times _K [S]^n \to [G\times _K S]^n$ such that $\hat \imath \circ \jmath = \bar \imath$ where $\jmath \;:\; [S]^n \to G\times _K[S]^n$ . As a consequence of the fourth statement, $\imath \;:\; [S]^n \to [G\times _K S]^n$ induces an injective map $[S]^n/K \to [G\times _K S]^n/G$ and $K_A = G_{\imath (A)}$ for every $A\in [S]^n$ ; therefore, $\hat \imath$ is injective.

As the reader might have noticed, this construction is an instance of general categorical phenomena; that formulation will have no use in this article so we will not discuss it.

4.3. Graphs as 1-dimensional complexes

While the objectives of this section only require us to consider simplicial graphs, the category of simplicial graphs does not have pushouts [Reference Stallings20]. For this reason, it is convenient to work within the framework of one-dimensional complexes or equivalently graphs in the sense that we describe below. We will only consider a particular class of pushouts of graphs that behaves well over simplicial graphs. A graph is a triple $(V,E,r)$ , where $V$ and $E$ are sets, and $r\;:\; E \to [V]^2$ is a function where $[V]^n$ is the collection of nonempty subsets of $V$ of cardinality at most $n$ . Elements of the set $V$ and $E$ are called vertices and edges, respectively; the function $r$ is called the attaching map. For a graph $\Gamma$ , we denote $V(\Gamma )$ and $E(\Gamma )$ its vertex and edge set, respectively. If $v\in V(\Gamma )$ , $e\in E(\Gamma )$ and $v\in r(e)$ , then $v$ is incident to $e$ , and $v$ is called an endpoint of $e$ . Vertices incident to the same edge are called adjacent.

The graph $(V,E,r)$ is simplicial if every edge has two distinct endpoints and $r$ is injective. Equivalently, $(V,E,r)$ is simplicial if $r\;:\; E \to [V]^2$ is injective and its image does not intersect $[V]^1$ .

A graph $\Delta$ is a subgraph of a graph $\Gamma$ if $V(\Delta ) \subset V(\Gamma )$ , $E(\Delta ) \subset E(\Gamma )$ and $r_\Delta$ equals the restriction of $r_\Gamma$ to $E(\Delta )$ . Abusing notation, we consider any vertex of a graph $\Gamma$ as an edgeless subgraph with a single vertex, and any edge $e$ of $\Gamma$ as the subgraph with vertex set the set of vertices incident to $e$ in $\Gamma$ and edge set consisting of only $e$ .

For a vertex $u$ of a simplicial graph $\Gamma =(V,E,r)$ , let $\mathsf{star_\Gamma (u)}$ denote the subgraph with vertex set $V(\mathsf{star}(u))=\{u\}\cup \{v\in V\mid \text{$v$ is adjacent to $u$}\}$ and edge set $E(\mathsf{star}(u)) = \{e\in E\mid \text{the endpoints of $e$ belong to $V(\mathsf{star}(u))$}\}$ and the attaching map is the corresponding restriction of $r$ .

Our notion of morphism allows the collapse of edges to single vertices. Specifically, a morphism of $\phi \;:\; (V,E,r) \to (V',E',r')$ of graphs is a pair of maps $\phi _0\;:\; V\to V'$ and $\phi _1\;:\; E \to V'\cup E'$ such that there is a commutative diagram

where the horizontal bottom arrow $\phi _0$ is the natural $G$ -map induced by $\phi _0\;:\; V\to V'$ , and $V'\to [V']^1$ is the natural bijection given by $v\mapsto \{v\}$ . Observe that in general for a morphism $\phi =(\phi _0,\phi _1) \;:\; \Gamma \to \Delta$ of graphs, the map $\phi _0$ does not determine $\phi _1$ ; however if $\Delta$ is simplicial then $\phi _0$ determines $\phi _1$ . A morphism $(\phi _0,\phi _1)$ is a monomorphism (also called an embedding) if both maps are injective.

Given a graph morphism $\phi =(\phi _0, \phi _1)\;:\; \Gamma \to \Delta$ and a subgraph $\Theta$ of $\Delta$ , the preimage $\phi ^{-1}(\Theta )$ is the subgraph of $\Gamma$ with vertex set $\phi _0^{-1}(V(\Theta ))$ and edge set $\phi _1^{-1}(V(\Theta )\cup E(\Theta ))$ .

Let $G$ be a group. A $G$ -graph is a graph $(V,E,r)$ where $V$ and $E$ are $G$ -sets, and $r$ is a $G$ -map with respect to the natural $G$ -action on $[V]^2$ induced by the $G$ -set $V$ . A morphism $(\phi _0,\phi _1)$ of $G$ -graphs is a morphism of graphs such that each $\phi _i$ is a $G$ -map. A $G$ -equivariant embedding is a monomorphisms of $G$ -graphs. A $G$ -action on a graph $\Gamma$ has no inversions if for every $e\in E$ and $g\in G$ such that $g.e=e$ , $g.v=v$ for every $v\in r(e)$ . For a $G$ -action without inversions on a graph $\Gamma$ and $K\leq G$ , let $\Gamma ^K$ denote subgraph of $\Gamma$ defined by $V(\Gamma ^K) =\{v\in V(\Gamma )\mid k.v=v \text{ for all $k\in K$}\}$ and $E(\Gamma ^K) =\{e\in E(\Gamma )\mid k.e=e \text{ for all $e\in K$}\}$ .

4.4. Extending group actions on graphs

Let $K$ be a subgroup of $G$ , and let $\Lambda =(V,E,r)$ be a $K$ -graph. Define

\begin{equation*} G\times _K \Lambda = (G\times _K V, G\times _K E, \tilde r)\end{equation*}

where $\tilde r$ is unique $G$ -map induced by the commutative diagram

where $\imath \;:\; V \hookrightarrow G\times _K V$ and $\jmath \;:\; E \hookrightarrow G\times _K E$ are the canonical $K$ -maps, see Lemma 4.2(3). Note that there is a canonical $K$ -equivariant embedding

\begin{equation*} \Lambda \hookrightarrow G\times _K \Lambda \end{equation*}

induced by $\imath$ and $\jmath$ . We consider $\Lambda$ a $K$ -subgraph of $G\times _K \Lambda$ .

4.5. Pushouts of graphs

Let $\mathsf X$ and $\mathsf Y$ be $G$ -graphs, let $C\leq G$ be a subgroup and suppose $\mathsf X^C$ and $\mathsf Y^C$ are nonempty. Let $x\in \mathsf X^C$ and $y\in \mathsf Y^C$ be vertices. The $C$ -pushout $\mathsf Z$ of $\mathsf X$ and $\mathsf Y$ with respect to the pair $(x,y)$ is the $G$ -graph $\mathsf Z$ obtained by taking the disjoint union of $\mathsf X$ and $\mathsf Y$ and then identifying the vertex $g.x$ with the vertex $g.y$ for every $g\in G$ .

Equivalently, the $C$ -pushout $\mathsf Z$ of $\mathsf X$ and $\mathsf Y$ with respect to the pair $(x,y)$ is the $G$ -graph $\mathsf Z$ whose vertex set $V(Z)$ is the pushout of the $G$ -maps $\kappa _1\;:\; G/C\to \mathsf V(\mathsf X)$ and $\kappa _2\;:\; G/C\to \mathsf V(\mathsf Y)$ given by $C\mapsto x$ and $C\mapsto y$ ; and edge set the disjoint union of the $G$ -sets $E(\mathsf X)$ and $E(\mathsf Y)$ , and attaching map $E(\mathsf Z) \to V(\mathsf Z)^2$ defined as the union of the attaching maps for $\mathsf X$ and $\mathsf Y$ postcomposed with the maps $V(\mathsf X) \to V(\mathsf Z)$ and $V(\mathsf Y) \to V(\mathsf Z)$ defining the pushout.

The standard universal property of pushouts holds for this construction: if $\jmath _1\;:\; \mathsf X \to \mathsf W$ and $\jmath _2\;:\; \mathsf Y \to \mathsf W$ are morphisms of $G$ -graphs such that $\jmath _1\circ \kappa _1 = \jmath _2\circ \kappa _2$ , then there is a unique morphism of $G$ -graphs $\mathsf Z\to \mathsf W$ such that above diagram commutes.

Example 4.2. Let $A=\langle a_1,a_2,a_3\mid [a_1,a_2] \rangle$ and $B=\langle b_1,b_2,b_3\mid [b_1,b_2] \rangle$ , and let $\mathsf X=\hat \Gamma (A,\langle a_1,a_2 \rangle, a_3)$ and $\mathsf Y=\hat \Gamma (B,\langle b_1,b_2 \rangle, b_3)$ be the coned-off Cayley graphs. Note that $\mathsf X$ is the Bass–Serre tree of the splitting of $A$ as the graph of groups

with two vertices and two edges with trivial edge group.

Let $G=A\ast _C B$ be the amalgamated product where $C$ corresponds to the cyclic subgroup $\langle a_1 \rangle \leq A$ and $\langle b_1 \rangle \leq B$ . Consider the $C$ -pushout $\mathsf Z$ of $G\times _A \mathsf X$ and $G\times _B \mathsf Y$ . By the fourth, fifth and sixth statements of Proposition 4.5 below, $\mathsf Z$ is a tree, it contains three distinct $G$ -orbits of vertices, two of these $G$ -orbits have all representatives with trivial stabilizer, and there is a vertex $z$ with stabilizer $\langle a_1, a_2, b_2\rangle = \langle a_1, a_2\rangle \ast _{\langle a_1 \rangle =\langle b_1 \rangle } \langle b_1, b_2 \rangle$ , and there are four distinct orbits of edges all with representatives having trivial stabilizer. Hence, $\mathsf Z$ is the Bass-Serre tree of a splitting of $G$ given by the graph of groups

with three vertices and four edges. In particular, $Z$ is the coned-off Cayley graph of $\hat \Gamma (G, G_y, \{a_3,b_3\})$ .

Proof. The first item is a direct consequence of Proposition 4.1. For the second item, first note that that the composition $\mathsf X\hookrightarrow G\times _A \mathsf X \xrightarrow{\imath _{1}} \mathsf Z$ is a morphism of $A$ -graphs. Observe that to prove the embedding part is enough to consider only vertices of $\mathsf X$ that are in the $A$ -orbit of $x$ . Suppose that $a.x$ and $x$ with $a\in A$ both map to $z \in \mathsf Z$ . Then, $a \in G_{z} = A_{x} \ast _C B_y$ and therefore $a\in A_x$ and hence $a.x=x$ . Item three follows directly from the definition of $\mathsf Z$ , and items four to six are consequences of Proposition 4.2.

To prove the seventh statement suppose that $X$ and $Y$ are connected graphs. The subgraph $\mathsf X\cup \mathsf Y$ of $\mathsf Z$ is connected since both $\mathsf X$ and $\mathsf Y$ contain the vertex $z$ . On the other hand, any vertex of $\mathsf Z$ belongs to a translate of $\mathsf X \cup \mathsf Y$ by an element of $G$ . Therefore to prove that $\mathsf Z$ is connected, it is enough to show that for any $g\in G$ there is a path in $\mathsf Z$ from $z$ to $g.z$ . For any $g\in G$ and $a\in A$ , there is a path from $g.z$ to $ga.z$ in $\mathsf Z$ : indeed, there is a path from $z$ to $a.z$ in the connected $A$ -subgraph $\mathsf X$ of $\mathsf Z$ , and hence there is a path from $g.z$ to $ga.z$ in $\mathsf Z$ . Analogously, for any $g\in G$ and $b\in B$ , there is a path from $g.z$ to $gb.z$ . Since any element of $G$ is of the form $a_1b_1\ldots a_nb_n$ with $a_i\in A$ and $b_i \in B$ , there is a path from $z$ to $g.z$ for any $g\in G$ .

Now we prove the eighth statement. Observe that $G$ splits as $G=A\ast _{A_x}(A_x\ast _CB_y)\ast _{B_y}B$ where the subgroups $A_x$ , $B_y$ and $A_x\ast _C B_y$ are naturally identified with the $G$ -stabilizers of $x\in G\times _A \mathsf X$ , $y\in G\times _B \mathsf Y$ , and $z\in \mathsf Z$ . Let $T$ denote the Bass–Serre tree of this splitting. The vertex and edge sets of $T$ can be described as:

\begin{equation*} V(T) = G/A \sqcup G/(A_x\ast _C B_y) \sqcup G/B \end{equation*}

and

\begin{equation*} E(T) = \{ \{ gA, g(A_x\ast _C B_y) \} \mid g\in G \} \sqcup \{ \{g(A_x\ast _C B_y), gB\} \mid g\in G \}\end{equation*}

respectively. Note that $T$ is a bipartite $G$ -graph, the equivariant bipartition of the vertices given by $G/A\sqcup G/B$ and $G/(A_x\ast _CB_y)$ .

Consider the $A$ -map from $\mathsf X$ to $T$ that maps every vertex of $\mathsf X$ not in the $A$ -orbit of $x$ to the vertex $A$ , and $x\mapsto A_x\ast _C B_y$ . Since $T$ is simplicial, this induces a unique morphism of $G$ -graphs $\jmath _1\;:\; G\times _A \mathsf X \to T$ . Analogously, there is $B$ -map $\mathsf Y\to T$ that maps every vertex not in the $B$ -orbit of $y$ to the vertex $B$ and $y\mapsto A_x\ast _C B_y$ ; this induces a unique $G$ -map $\jmath _2\;:\; G\times _B \mathsf Y \to T$ .

Consider the $G$ -maps $\kappa _1 \;:\; G/C \to G\times _A \mathsf X$ and $\kappa _2\;:\; G/C \to G\times _B \mathsf Y$ given by $C\mapsto x$ and $C\mapsto y$ , respectively. Since $\jmath _1\circ \kappa _1 = \jmath _2\circ \kappa _2$ , the universal property implies that there is a surjective $G$ -map $\xi \;:\; \mathsf Z\to T$ .

Note that $\xi ^{-1}(\xi (z))$ is contained in the orbit $G.z$ . Suppose $g.z\in \xi ^{-1}(\xi (z))$ . Then $g(A_x\ast _C B_y) = A_x\ast _C B_y$ and hence $g\in A_x\ast _C B_y$ . Since $A_x\ast _C B_y$ is the $G$ -stabilizer of $z$ , we have that $g.z=z$ . This shows that $\xi ^{-1}(\xi (z))=\{z\}$ .

Let us conclude by proving that if $v\in V(T)$ is not in the $G$ -orbit of $\xi (z)=A_x\ast _c B_y$ then $\xi ^{-1}(\mathsf{star_T}(v))$ is a graph isomorphic to either $\mathsf X$ or $\mathsf{Y}$ . Note that such a vertex $v$ is an element of $G/A \cup G/B$ . By equivariance, it is enough to consider the two symmetric cases, namely $v=A$ or $v=B$ . Let us prove that $\xi ^{-1}(\mathsf{star}_T{A})$ is isomorphic to $\mathsf X$ . Observe that any edge of $\mathsf{star}_T(A)$ is of the form $\{A, a(A_x\ast _C B_y)\}$ with $a\in A$ . Since $(\xi \circ \imath _1)^{-1}(\mathsf{star}_T(A)) = \jmath _1^{-1}(\mathsf{star}_T(A))$ and $\xi ^{-1}(\mathsf{star}_T(A)) \subset \imath _1(G\times _A X)$ , we have that $\xi ^{-1}(\mathsf{star}_T(A)) = \imath _1(\jmath _1^{-1}(\mathsf{star}_T(A))$ . Recall that the canonical $A$ -map $\mathsf{X} \to G\times _A \mathsf{X}$ is injective, and this defines a natural identification of $\mathsf{X}$ with a subgraph of $G\times _A \mathsf{X}$ which equals $\jmath ^{-1}(\mathsf{star}_T(A))$ by definition of $\jmath$ . Then we have that $\imath _1(\jmath _1^{-1}(\mathsf{star}_T(A))=\imath _1(\mathsf{X})$ is isomorphic to $\mathsf X$ by the second item of the proposition.

4.6. Proof of Theorem 3.1

Proof. Note for any vertex $u$ of $\Gamma$ , if $\xi (u)\in L$ then $u$ is a cut vertex of $\Gamma$ . The bipartite assumption on $T$ implies that if $\Theta$ is the closure of a connected component of $\Gamma \setminus \xi ^{-1}(L)$ , then $\Theta$ equals some $\Delta \in \Omega$ .

The first and second statements follow from the previous observation, the second one with important generalizations [Reference Bestvina and Feighn3]. For the third statement, if $\Gamma$ is fine at $u$ , then any subgraph containing $u$ is fine at $u$ . Conversely, let $u$ be a vertex of $\Gamma$ such that any $\Delta$ containing $u$ is fine at $u$ . There are two cases to consider.