From Free Splittings To Graphs Of Groups

06 Jan 2022

In the previous post, we defined free splittings for a group \(F\). The quotient of a free splitting \(T\) of \(F\) by the action of \(F\) is a graph of groups with trivial edge groups. The purpose of this post is to introduce maps of graphs of groups with trivial edge groups and explore lifting maps of the quotient graph of groups to twisted equivariant maps of the tree \(T\).

Graphs of groups with trivial edge groups

Let \(G\) be a finite graph. The structure of a graph of groups with trivial edge groups on \(G\) is the assignment, for each vertex \(v\) of \(G\), of a group \(\mathcal{G}_v\). We write \(\mathcal{G}\) for the graph of groups. Associated to \(\mathcal{G}\) we can form a graph of spaces \(X_{\mathcal{G}}\) à la Scott and Wall. For each vertex \(v\) of \(G\), take a CW complex \(X_v\) with one vertex \(\star_v\) that is a \(K(\mathcal{G}_v,1)\). The space \(X_{\mathcal{G}}\) is formed from the disjoint union of these CW complexes by adding, for each edge \(e\) of \(G\) with initial vertex \(v\) and terminal vertex \(w\), an edge connecting \(\star_v\) to \(\star_w\). There is a natural “retraction” \(r\colon X_{\mathcal{G}} \to G\) that collapses each \(K(\mathcal{G}_v,1)\) to the vertex \(\star_v\).

The fundamental group of the graph of groups \(\mathcal{G}\) is the fundamental group of the graph of spaces \(X_{\mathcal{G}}\). If we choose a basepoint \(p \in X_{\mathcal{G}}\) in the image of the retraction, we may homotope each element of \(\pi_1(X_{\mathcal{G}},p)\) into the \(1\)-skeleton, and further to an edge path in \(\mathcal{G}\) of the form

\[\gamma = e'_1g_1e_2g_2\ldots e_kg_ke'_{k+1},\]

where \(e'_1\) and \(e'_{k+1}\) are terminal and initial segments of oriented edges \(e_1\) and \(e_{k+1}\) of \(G\), respectively, where \(e_2,\ldots,e_k\) are oriented edges of \(G\), where \(g_i\) for \(1 \le i \le k\) is an element of \(\mathcal{G}_{v_i} = \pi_1(X_{v_i},\star_{v_i})\), and where \(v_i\) is the terminal vertex of the edge \(e_i\) and the initial vertex of the edge \(e_{i+1}\). Notice that this notion makes sense without reference to \(X_{\mathcal{G}}\). Homotopy rel endpoints for edge paths in \(\mathcal{G}\) has a combinatorial flavor that makes no reference to \(X_{\mathcal{G}}\): it is generated by performing multiplication or factoring in vertex groups and adding or removing subpaths of the form \(e \bar e\), where \(\bar e\) is the edge \(e\) in its opposite orientation. An edge path is tight if the number of edges in the path cannot be decreased by a homotopy. Every edge path is homotopic to a unique tight path (which may contain no edges).

Bass–Serre theory asserts that \(F = \pi_1(\mathcal{G},p) = \pi_1(X_{\mathcal{G}},p)\) is a free product of the form

\[F = A_1 * \cdots * A_n * F_k,\]

Where the \(A_i\) are the nontrivial vertex groups of \(\mathcal{G}\) and \(F_k\), a free group of rank \(k\), is the ordinary fundamental group of \(G\).

The Bass–Serre tree

Given a graph of groups with trivial edge groups \(\mathcal{G}\) and a point \(p \in G\), we wish to define a free splitting \(\Gamma\) of \(F\) equipped with a basepoint \(\tilde p\) called the Bass–Serre tree of \(\mathcal{G}\). Given a vertex \(v\) of \(G\), write \([p,v]\) for the set of homotopy classes of edge paths in \(\mathcal{G}\) beginning at \(p\) and ending at \(v\). There is a natural right action of \(\mathcal{G}_v\) on \([p,v]\) defined by \(\gamma.g = \gamma g\). Let \(V\) be the set of vertices of \(G\). The vertex set of \(\Gamma\) is the set

\[\coprod_{v \in V}[p,v]/\mathcal{G}_v.\]

Two vertices \([\gamma] \mathcal{G}_v\) and \([\gamma']\mathcal{G}_w\) are adjacent in \(\Gamma\) if and only if the path \(\bar\gamma\gamma'\) is homotopic to a path of the form \(g e g'\), i.e. a path of length one. One can check that this is well-defined independent of the choice of representatives \(\gamma\) and \(\gamma'\). There is a natural action of \(F = \pi_1(\mathcal{G},p)\) on \(\Gamma\) defined as \([\gamma].[\sigma]\mathcal{G}_v = [\gamma\sigma]\mathcal{G}_v\). The fundamental theorem of Bass–Serre theory asserts that \(\Gamma\) is a tree, that the \(F\)-action has trivial edge stabilizers, and that the “quotient graph of groups” \(F\backslash\backslash\Gamma\), which I won’t describe here, is naturally identified with \(\mathcal{G}\). Let \(\pi\colon \Gamma \to G\) be the natural projection.

Suppose \(\gamma\) is a tight path in \(\mathcal{G}\) beginning at a vertex \(v\) and ending at a vertex \(w\). A lift \(\tilde v\) of \(v\) to \(\Gamma\) is determined by a tight path \(\sigma\) from \(p\) to \(v\). The path \(\gamma\) lifts uniquely to a tight path \(\tilde\gamma\) in \(\Gamma\) beginning at \(\tilde v\): it is the unique tight path connecting the vertex \([\sigma]\mathcal{G}_v\) to the vertex \([\sigma\gamma]\mathcal{G}_w\). More generally it is possible to lift edge paths that are not tight or do not begin and end at vertices. In particular lifting the trivial path at \(p\) defines a basepoint \(\tilde p\) in \(\Gamma\).

Maps of graphs of groups with trivial edge groups

Let \(\mathcal{G}\) be a graph of groups with trivial edge groups and \(X_{\mathcal{G}}\) an associated graph of spaces. A map of graphs of groups with trivial edge groups \(f\colon \mathcal{G} \to \mathcal{G}\) is a pair of maps \(f_X\colon X_{\mathcal{G}} \to X_{\mathcal{G}}\) and \(f\colon G \to G\) such that the following diagram commutes

\[\require{AMScd}\begin{CD} X_{\mathcal{G}} @>f_X>> X_{\mathcal{G}} \\ @VrVV @VVrV \\ G @>f>> G. \end{CD}\]

A homotopy of maps of graphs of groups with trivial edge groups is a pair of homotopies such that the relevant diagram commutes. A map \(f\colon \mathcal{G} \to \mathcal{G}\) is a homotopy equivalence if, as usual, there is a map \(g\colon \mathcal{G} \to \mathcal{G}\) such that \(gf\) and \(fg\) are each homotopic to the identity map.

It is not too hard to see that each map \(f\colon \mathcal{G} \to \mathcal{G}\) is homotopic to one that sends the vertices \(\star_v\) of \(X_{\mathcal{G}}\) to vertices of \(X_{\mathcal{G}}\) and further sends edges of \(G \subset X_{\mathcal{G}}\) to (possibly trivial) edge paths in \(\mathcal{G}\).

Like edge paths, a map of graphs of groups with trivial edge groups that is a homotopy equivalence has a combinatorial shadow in \(\mathcal{G}\) that does not depend on \(X_{\mathcal{G}}\): the data of \(f\) is a continuous map \(f\colon G \to G\) taking vertices to vertices and edges to edge paths, along with for each edge \(e\) of \(G\), an edge path \(f(e)\) in \(\mathcal{G}\), and for each vertex \(v\) of \(G\) with nontrivial vertex group, an isomorphism \(f_v\colon \mathcal{G}_v \to \mathcal{G}_{f(v)}\). Homotopy for maps of graphs of groups has a combinatorial shadow in \(\mathcal{G}\) as well; I won’t describe it.

Suppose \(f\colon \mathcal{G} \to \mathcal{G}\) is a homotopy equivalence of the form above. Given a basepoint \(p \in G\) and a path \(\sigma\) from \(p\) to \(f(p)\), the homotopy equivalence \(f\) induces an automorphism \(\Phi \colon \pi_1(\mathcal{G},p) \to \pi_1(\mathcal{G},p)\) by the rule

\[\Phi([\gamma]) = [\sigma f(\gamma) \bar\sigma].\]

If \(\Gamma\) is the Bass–Serre tree of \(\mathcal{G}\) with basepoint \(\tilde p\) lifting \(p\), the choice of path \(\sigma\) also defines a lift \(\tilde f\colon \Gamma \to \Gamma\) defined as follows: A point \(\tilde x\) of \(\Gamma\) is determined by an edge path \(\gamma\) in \(\mathcal{G}\) beginning at \(p\): if \(\tilde x\) is a vertex, the path \(\gamma\) is any path in the \(\mathcal{G}_x\)-orbit \([\gamma]\mathcal{G}_x\) determining \(\tilde x\). In this case, the point \(\tilde f(\tilde x)\) is \([\sigma f(\gamma)]\mathcal{G}_{f(x)}\). One checks that this is independent of the choice of \(\gamma\). This can be extended to define \(\tilde f\) for points that are not vertices. The map \(\tilde f\) is \(\Phi\)-twisted equivariant in the sense that

\[\tilde f(g.\tilde x) = \Phi(g).\tilde f(\tilde x)\]

for each point \(\tilde x \in \Gamma\) and element \(g\) of \(\pi_1(\mathcal{G},p)\). We say that \(\tilde f\) corresponds to \(\Phi\).

A little about the map \(D\tilde f\)

Continuing notation from the previous paragraph, let \(\tilde v\) be the vertex \([\gamma]\mathcal{G}_v\) of \(\Gamma\). Suppose the final vertex group element of \(\gamma\) is trivial. A lift \(\tilde f\colon \Gamma \to \Gamma\) of \(f\) determined by the choice of path \(\sigma\) from \(p\) to \(f(p)\) satisfies \(\tilde f(\tilde v) = \tilde v\) if and only if \(f\) fixes \(v\) and \(\sigma\) is homotopic to a path of the form \(\gamma h f(\bar\gamma)\) for \(h \in \mathcal{G}_v\). A direction at \(\tilde v\) is an edge \(\tilde e\) with initial vertex \(\tilde v\). The map \(g \mapsto [\gamma g\bar\gamma]\) determines an isomorphism from \(\mathcal{G}_v\) to the stabilizer of \(\tilde v\) in \(\pi_1(\mathcal{G},p)\). Under this isomorphism there is a (left) action of \(\mathcal{G}_v\) on the set of directions based at \(\tilde v\). Let \(\operatorname{st}(v)\) be the set of edges of \(G\) with initial vertex \(v\). There is a \(\mathcal{G}_v\)-equivariant bijection from the set of directions based at \(\tilde v\) with the set

\[\coprod_{e \in \operatorname{st}(v)} \mathcal{G}_v\times \{e\}\]

defined as follows. Let \(\tilde w\) be the terminal vertex of the edge \(\tilde e\). The map sends \(\tilde e\) to the pair \((g,e)\) if we have \(\tilde w = [\gamma ge]\mathcal{G}_w\). The map \(\tilde f\colon \Gamma \to \Gamma\) induces a map \(D\tilde f\) of the set of directions at \(\tilde v\) as follows: \(D\tilde f(\tilde e)\) is the first edge on the edge path \(\tilde f(\tilde e)\). We have

\[\tilde f(\tilde w) = [\sigma f(\gamma ge)]\mathcal{G}_{f(w)} = [\gamma h f(\bar\gamma)f(\gamma)f_v(g)f(e)]\mathcal{G}_{f(w)} = [\gamma h f_v(g) f(e)]\mathcal{G}{f(w)}.\]

If the edge path \(f(e)\) begins \(g_0 e_1\), we see that \(D\tilde f(g,e) = (hf_v(g)g_0,e_1)\).

My latest point of confusion

Suppose \(\tilde v\) has infinite valence, or equivalently that the group \(\mathcal{G}_v\) is infinite. Let’s assume \(\mathcal{G}_v\) is countable. To get the result that I want, I’m happy to restrict further, and I’m particularly interested in the case where \(\mathcal{G}_v\) is virtually finite-rank free.

Suppose \(\tilde f\) is a lift of \(f\colon \mathcal{G} \to \mathcal{G}\) that fixes \(\tilde v\). We say that \(\tilde f\) is principal if \(\operatorname{Fix}(\tilde f)\) contains at least two points or if \(D\tilde f\) fixes a direction based at \(\tilde v\). Two lifts \(\tilde f\) and \(\tilde f'\) fixing \(\tilde v\) corresponding to elements \(h\) and \(h'\) in \(\mathcal{G}_v\) are isogredient if there exists \(c \in \mathcal{G}_v\) such that the maps \(D\tilde f\) and \(D\tilde f'\) satisfy \(D\tilde f' = c D\tilde f c^{-1}\): that is, for a direction \((x,e)\), we have \((h' f_v(x) g_0,e_1) = (ch f_v(c^{-1}) f_v(x) g_0,e_1)\), which holds if and only if \(h' = ch f_v(c^{-1})\). It’s not too hard to see that if \((x,e)\) is a fixed direction for \(D \tilde f\), then \(cx\) is a fixed direction for \(D\tilde f'\), or more generally if \((x,e)\) is a periodic direction for \(D\tilde f\), then \(cx\) is a periodic direction for \(D\tilde f'\). We are interested in principal lifts up to isogredience, so we may simplify things by noting that up to isogredience, if there is a fixed point, we may assume that it is of the form \((1,e)\). In this case, we have

\[D\tilde f(1,e) = (hg_0,e),\]

so we conclude that \(h = g_0^{-1}\). Therefore up to isogredience, fixed directions are pretty simple. Periodic directions are a little more complicated: assuming that \(e_1 = e\), the condition that \((1,e)\) is mapped to itself by \(D\tilde f^k\) is that

\[h f_v(h)f_v^2(h)\ldots f_v^{k-1}(h)f_v^{k-1}(g_0)f_v^{k-2}(g_0) \ldots f_v(g_0)g_0 = 1.\]

What I want is sufficient conditions on \(\mathcal{G}_v\) to guarantee the existence of \(k > 0\) independent of the choice of lift \(\tilde f\) such that if \(D\tilde f^k\) has a periodic direction, then that direction is fixed. As already noted, we may simplify up to isogredience by assuming that the identity is periodic.