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Last time: Theorem 21 (Groves–Manning–Osin): If G is hyperbolic rel \mathcal P then there exists a finite subset A \subseteq G\setminus 1 such that if \bigcup_i N_i \cap A = \emptyset then
(a) P_i/N_i \to G/\mathcal N is injective;
(b) G is hyperbolic rel P_i/N_i.

Theorem 22 (Gromov, Olshanshkii, Delzant): If G is hyperbolic relative to the infinite cyclic \{\langle g_1\rangle,\dots,\langle g_n \rangle\} then there is a K>0 such that for all l_1,\dots,l_n>0 there exists a \phi : G \to G' hyperbolic such that o(\phi(g_i))=Kl_i for each i.

The proof is an easy application of Groves–Manning–Osin.

Definition: If \{\langle g_1 \rangle,\dots,\langle g_n\rangle\} (infinite cyclic) is malnormal then we say g_1,\dots,g_n are independent. A group G is omnipotent if for every independent g_1,\dots,g_n there exists a K>0 such that for all l_1,\dots,l_n>0 there exists a homomorphism $\phi$ from G to a finite group such that o(\phi(g_i)) = Kl_i for all i.

Omnipotence strengthens residual finiteness for torsionfree groups.

Exercise 29: If every hyperbolic group is residually finite then every hyperbolic group is omnipotent.

We’ll finish off by talking about a similar theorem of Agol–Groves–Manning. I’m going to seem a little cavalier about torsion. This is OK. In fact, if every hyperbolic group is residually finite then every hyperbolic group is virtually torsionfree.

Theorem 22 (Agol–Groves–Manning): If every hyperbolic group is residually finite then every quasi-convex subgroup H of any hyperbolic group G is separable.

Let g \in G \setminus H. The idea is to Dehn fill H to get a new hyperbolic group \bar G in which the image \bar H is finite and \bar g \not\in\bar H. If we could do this, we would be done by residual finiteness. This works if H is malnormal. But it probably isn’t. Fortunately, we can quantify how far H is from being malnormal:

Definition: The height of H is the maximal n \in \mathbb N such that there are distinct cosets g_1 H,\dots,g_n H \in G/H such that the intersection
g_1 H g_1^{-1} \cap \dots \cap g_n H g_n^{-1}
is infinite.

H is height 0 iff H is finite. In a torsionfree group, H is height 1 iff H is malnormal.

Theorem 23 (Gitik, Mitra, Rips, Sageev): A quasiconvex subgroup of a hyperbolic group has finite height.

Agol, Groves and Manning are able to prove:

Theorem 24: Let G be a (torsionfree) residually finite hyperbolic group, and H a quasiconvex subgroup of height k. Let g \in G\setminus H. Then is an epimorphism \eta: G \to \bar G to a hyperbolic group such that
(i) \eta(H) is quasiconvex in \bar G;
(ii) \eta(g) \not\in\eta(H);
(iii) \eta(H) has height \leq k-1.

The idea of the proof of Theorem 24 is to Dehn fill a finite index subgroup of a maximal infinite intersection of conjugates of H. Theorem 22 is an easy consequence.

We still need to convince ourselves of some basic facts about the previous lecture, for example is the map A \to A *_C B injective?

Example: Cut the sphere S^2 along the equator. Then the diagram we have is

Definition: If G=A *_C B or G = A*_C we say that G splits over C, and we call C the edge group. If G=A*_C or G = A *_C B and C is not A or B in the latter case, then we say that G splits non-trivially.

Definition: Let \Gamma be a connected graph (i.e. a 1-dimensional CW-complex). For each vertex v \in V(\Gamma) (resp. edge e \in E(\Gamma)) let G_v (resp. G_e) be a group. If v_\pm are vertices adjoining an edge e then let \partial_\pm^e : G_e \to G_{v_\pm} be an injective homomorphism. This data determines a graph of groups \mathcal G.

We say that \mathcal G has:

  • underlying graph \Gamma
  • vertex groups \{ G_v \}
  • edge groups \{ G_e \}
  • edge maps \{ \partial_\pm^e \}

Similarly, we have:

Definition: Let \Xi be a connected graph. For each vertex v \in V(\Xi) (resp. e \in E(\Xi)) let X_v (resp. X_e) be a connected CW-complex. If v_\pm adjoin e let \partial_\pm^e : X_e \to X_{v_\pm} be \pi_1-injective continuous maps. This data determines a graph of spaces \mathcal X. It has underlying graph \Xi, vertex spaces X_v, edge spaces X_e, etc. The graph of spaces \mathcal X determines a space as follows: define

\displaystyle X_{\mathcal X} = \left(\bigsqcup_{v \in V(\Xi)} X_v \sqcup \bigsqcup_{e\in E(\Xi)} (X_e \times [-1,+1])\right) / \sim,

where (x,\pm 1) \sim \partial_\pm^e(x) for x \in X_e. We say that \mathcal{X} is a graph-of-spaces structure (or decomposition) for X_\mathcal{X}.

Remark: There is a natural map X_\mathcal{X} \to \Xi (by collapsing all the edge and vertex spaces).

Given any graph of groups \mathcal G we can construct a graph of spaces \mathcal X with underlying graph \Gamma by assigning X_v = K(G_v,1), X_e = K(G_e,1) and realizing the edge maps as continuous maps X_e \to X_{v_\pm}. We write X_\mathcal G for X_\mathcal X. This is well-defined up to homotopy equivalence.

Definition: The fundamental group of \mathcal G is just \pi_1(\mathcal G) = \pi_1(X_\mathcal{G}).

Examples:

  • If then \pi_1 (G) = A *_C B
  • If One vertex labeled A, one edge labeled C then \pi_1 (G) = A *_C
  • Let \mathcal C \subset \Sigma be an embedded multicurve (disjoint union of circles) inside a surface. Cutting along \mathcal C decomposes \Sigma into a graph of spaces and \pi_1 (\Sigma) into a graph of groups.

Note: The edge maps of X_\mathcal{G} are only defined up to free (i.e. unbased) homotopy. Translated to \mathcal G, this means that only the conjugacy class of \partial_\pm in G_{v_\pm} matters.

Remark: The map X_\mathcal G \to \Gamma induces a surjection \pi_1(\mathcal G) \to \pi_1(\Gamma).

Here’s a way to construct a graph of groups. Let’s suppose G acts on a tree T without edge inversions (we can do this by subdividing edges if necessary). Let Y = \widetilde{K(G,1)}. The group G acts diagonally on T \times Y. The quotient X = G\setminus (T \times Y) has a structure of a graph of spaces. The underlying graph is \Gamma = G\setminus T and there is a natural map X \to \Gamma.

Let v \in V(\Gamma) be a vertex below \tilde v \in V(T). The preimage of v is just v \times (G_{\tilde v} \setminus Y) where G_{\tilde v} is the stabilizer of \tilde v. Similarly, for e \in E(\Gamma) below \tilde e \in E(T), the preimage of e is e \times (G_{\tilde e} \setminus Y).

If \tilde e adjoins \tilde v then G_{\tilde e} \subset G_{\tilde v} so the edge map G_{\tilde e} \setminus Y \to G_{\tilde v} \setminus Y is a covering map and therefore \pi_1-injective. We have defined a graph of spaces \mathcal X and \pi_1(X_\mathcal X) = G since T \times Y is simply connected.

Applying \pi_1 to everything, we have a graph of groups \mathcal G. Its underlying graph is \Gamma. Its vertex groups are the vertex stabilizers of T, its edge groups are the edge stabilizers, and the edge maps are the inclusions. Also, \pi_1(\mathcal G) = G.

Question for next time: Does every graph of groups arise in this way?

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