Shelters In Graphs Of Groups

26 Mar 2022

Let’s fix a group \(G\) that acts cocompactly on a tree \(T\). Forester introduced the notion of a deformation of the tree \(T\) and Guirardel and Levitt studied the deformation space \(\mathscr{D}\) consisting of trees obtainable from \(T\) by a finite sequence of deformations. Usually we assume \(G\) to be finitely generated, but it’s not clear to me how often this assumption is necessary. Maybe that’ll be a topic for another blog post. In this post I want to discuss the notion of a shelter, which Guirardel and Levitt (and also Clay) use to construct a deformation retraction of \(\mathscr{D}\) (or its simplicial spine) onto a smaller, often finite-dimensional, space.

The quotient graph of groups

Given a group \(G\) acting on a graph \(\Gamma\), there is a notion of a quotient graph of groups, which I’d like to describe once and for all here. Let me remind you that a graph of groups is a graph, which I’ll call \(\mathcal{G}\), together with an assignment of groups \(\mathcal{G}_v\) and \(\mathcal{G}_e\) to each vertex \(v\) and edge \(e\) of \(\mathcal{G}\), along with injective homomorphisms \(\iota_e \colon \mathcal{G}_e \to \mathcal{G}_v\) whenever \(v\) is the terminal vertex of the oriented edge \(e\).

Anyway, suppose we have \(G\) acting on \(\Gamma\). I want this action to be essentially combinatorial, in the sense that we can think of \(\Gamma\) as having a set of vertices and a set of edges, and \(G\) permutes the elements of those sets preserving adjacency. Making this precise is surprisingly exhausting (“fix a homeomorphism between each closed edge…”), so I won’t. By \(G\)-equivariantly subdividing certain orbits of edges, we may suppose that the action is without inversions in edges, which means that if a group element preserves an edge \(e\) of \(\Gamma\), then it fixes each vertex incident to \(e\). The advantage of this condition is that the quotient \(G\backslash\Gamma\) then inherits the structure of a graph, call it \(\mathcal{G}\). To make \(\mathcal{G}\) into a graph of groups, we need groups and homomorphisms.

So fix connected subgraphs \(S_0 \subset S_1 \subset \Gamma\) with the property that the natural projection \(p\colon \Gamma \to \mathcal{G}\) restricts to a bijection on the set of vertices of \(S_0\) and a bijection on the set of edges of \(S_1\). In other words, \(S_1\) is a fundamental domain for the action of \(G\) on \(\Gamma\), and if we assume \(S_0\) is a minimal subgraph of \(S_1\) with the defining property, it follows that \(S_0\) projects to a spanning tree of \(\mathcal{G}\). For a vertex \(v\) and an edge \(e\) of \(\mathcal{G}\), write \(\tilde v\) and \(\tilde e\) for its unique preimage in \(S_0\) and \(S_1\) respectively. For each oriented edge \(e\) of \(\mathcal{G}\), choose a group element \(g_e \in G\) with the following property: if \(x\) is the terminal vertex of \(\tilde e\) and \(p(x) = v\), we have that \(g_e.x = \tilde v\). Furthermore, if \(x = \tilde v\), choose \(g_e = 1\). The groups \(\mathcal{G}_v\) and \(\mathcal{G}_e\) are the stabilizers in \(G\) of \(\tilde v\) and \(\tilde e\) respectively, and the injective homomorphism \(\iota_e \colon \mathcal{G}_e \to \mathcal{G}_v\) is the restriction of the map \(h \mapsto g_e h g_e^{-1}\) to \(\mathcal{G}_e\).

Properties of the tree in the quotient

Let’s return to the situation of our group \(G\) acting cocompactly on a tree \(T\). Then the quotient graph of groups \(\mathcal{G}\) is finite and connected. We assume the action is minimal, in the sense that there is no \(G\)-invariant subtree. In the quotient graph of groups \(\mathcal{G}\), this is equivalent to the condition that no valence-one vertex \(v\) has the property that the incident edge-to-vertex group inclusion is surjective. We also assume that \(T\) has no subdivision vertices, i.e. that a valence-two vertex of \(T\) has the property that the incident edges are in the same \(G\)-orbit. In the quotient graph of groups \(\mathcal{G}\), this is equivalent to the condition that no valence-two vertex \(v\) has the property that both incident edge-to-vertex group inclusions are surjective.

A general fact about automorphisms of simplicial trees (in fact, isometries of real trees) is that each automorphism (isometry) is either elliptic, i.e. fixes a point of the tree or hyperbolic i.e. there is an embedded copy of \(\mathbb{R}\) called the axis in the tree that is preserved by the automorphism (isometry) and along which the automorphism (isometry) acts by translation.

An edge \(\tilde e\) of \(T\) is collapsible if collapsing each edge in its orbit to a point does not change the partition of elements of \(G\) into hyperbolic and elliptic elements. Equivalently, the \(G\)-orbit of \(\tilde e\) must not contain the axis of a hyperbolic element. In the quotient graph of groups, we must have that the endpoints of \(p(\tilde e) = e\) are distinct and that at least one of the edge-to-vertex group inclusions is surjective. If \(\tilde e\) is collapsible, collapsing each edge in its orbit to a point yields a new tree \(T'\) in the same deformation space as \(T\). In my paper, I describe this as a collapse map of the quotient graphs of groups which is a homotopy equivalence.

A tree \(T\) is reduced if is has no collapsible edges. In the quotient graph of groups, this says that if some edge-to-vertex group inclusion \(\iota_e\colon \mathcal{G}_e \to \mathcal{G}_v\) is surjective, then the edge \(e\) forms a loop. Every cocompact \(G\)-tree \(T\) may be made reduced by a finite sequence of collapse maps.

Shelters

An edge \(\tilde e\) in a tree \(T\) is surviving if there is a reduced tree \(T'\) and a product of collapse maps \(T \to T'\) with the property that the edge \(\tilde e\) is not collapsed. The deformation retraction mentioned in the first paragraph consists of those trees all of whose edges are surviving. For the reader familiar with Culler–Vogtmann Outer Space, it’s not hard to see that a tree in Outer Space has the property that every edge is surviving if and only if the quotient graph has no separating edges. The concept of a shelter generalizes this “no separating edges” condition.

Let \(\mathcal{G}\) be the quotient graph of groups. Label each half-edge of \(\mathcal{G}\) by \(=\) or \(\neq\) according to whether the corresponding edge-to-vertex group inclusion is surjective or non-surjective, respectively. An embedded segment or circle \(\gamma\) in \(\mathcal{G}\) is a shelter if it satisfies one of the following properties:

  1. \(\gamma\) is a segment with \(\ne\) at each of its endpoints and \(=\) otherwise.

  2. \(\gamma\) is a circle with two \(=\) labels at all of its vertices with the possible exception of one.

  3. \(\gamma\) is a circle, and there is an orientation of \(\gamma\) such that all positively oriented edges have \(=\) at their endpoints.

Clay has a nice picture as Figure 1 in his paper. Clay proves the following as Proposition 1.13 of his paper.

Theorem. An edge \(\tilde e\) of \(T\) is surviving if and only if \(p(\tilde e)\) is contained in a shelter.