392C Geometric Group Theory

29. Doubles of free groups

8 April 2009 in Course notes | Tags: Doubles of free groups, Elevations, Separability | by yyao392c | 1 comment

Agol-Groves-Manning’s Theorem predicts that, for every word-hyperbolic group we can easily construct, every quasiconvex subgroup is separable (otherwise, we would find a non-residually finite hyperbolic group!).

In this section, we use graphs of groups to build new hyperbolic groups:

Combination Theorem (Bestvina & Feighn): If $H$ is a quasiconvex malnormal subgroup of hyperbolic groups $G_1, G_2$ , then $G_1\ast_H G_2$ is hyperbolic.

Recall: $H$ is called a malnormal subgroup of $G$ if it satisfies: if $gHg^{-1}\cap H\neq 1$ , then $g\in H$ .

For a proof, see M. Bestvina and M. Feighn, “A combination theorem for negatively curved groups”, J. Differential Geom., 35 (1992), 85–101.

Example: Let $F$ be free, $w\in F$ not a proper power. By Lemma 11, $\langle w\rangle\leq F$ is malnormal, so $D:=F\ast_{\langle w\rangle} F$ is hyperbolic. As a special case, if $\Sigma$ is closed surface of even genus $n=2k$ , considered as the connected sum of two copies of the closed surface of genus $k$ , then by Seifert-van Kampen Theorem, $\pi_1(\Sigma)=F_{2k}\ast_{\langle w\rangle} F_{2k}$ for some $w\in F_{2k}$ .

Question: (a) Which subgroups of $D$ are quasiconvex? (b) Which subgroups of $D$ are separable?

We will start by trying to answer (b). The following is an outline of the argument: Let $\Gamma$ be a finite graph so that $\pi_1(\Gamma)=F$ , let $\Gamma_{\pm}$ be two copies of $\Gamma$ . Realize $w\in F$ as maps $\partial^{\pm}: C\rightarrow \Gamma_{\pm}$ , where $C\simeq S^1$ . Let $X$ be the graph of spaces with vertex spaces $\Gamma_{\pm}$ , edge space $C$ , and edge maps $\partial^{\pm}$ . Then clearly, $D\simeq \pi_1(X)$ , and finitely generated subgroups $H\leq D$ are in correspondence with covering spaces $X^H\rightarrow X$ . We can then use similar technique to sections 27 and 28.

Let us now make a few remarks about elevations of loops. Let $f: C\rightarrow X$ be a loop in some space $X$ , i.e., $C\simeq S^1$ and $\pi_1(C)\simeq\mathbf{Z}$ . Consider an elevation of $f$ :

diagram

The conjugacy classes of subgroups of $\mathbf{Z}$ are naturally in bijection with $\mathbf{N}\cup\{\infty\}$ . The degree of the elevation is equal to the degree of the covering map $C'\rightarrow C$ .

Definition: Suppose $X'\rightarrow X$ is a covering map and $\widehat{X}$ is an intermediate covering space, i.e., $X'\rightarrow X$ factors through $\widehat{X}\rightarrow X$ , and we have a diagram

diagram1

If $f'$ and $\widehat{f}$ are elevations of $f$ and the diagram commutes, then we say that $f'$ descends to $\widehat{f}$ .

Let $\Gamma$ be a finite graph, $H\leq \pi_1(\Gamma)$ a finitely generated subgroup and $f: C\rightarrow\Gamma$ a loop. Let $\Gamma^H\rightarrow\Gamma$ be a covering space corresponding to $H$ .

Lemma 29: Consider a finite collection of elevations $\{f'_i: C'_i\rightarrow\Gamma^H\}$ of $f$ to $\Gamma^H$ , each of infinite degree. Let $\Delta\leq\Gamma^H$ be compact. Then for all sufficiently large $d>0$ , there exists an intermediate, finite-sheeted covering space $\Gamma_d\rightarrow\Gamma$ satisfying: (a) $\Delta$ embeds in $\Gamma_d$ ; (b) every $f'_i$ descends to an elevation $\widehat{f}_i: \widehat{C}_i\rightarrow\Gamma_d$ of degree exactly $d$ ; (c) these $\widehat{f}_i$ are pairwise distinct.

28. Separability properties of amalgams

8 April 2009 in Course notes | Tags: Graphs of groups, Residual finiteness | by drosen2000 | Leave a comment

As before, are $\chi'$ and $\chi$ are graphs of spaces equipped with maps $\Phi \colon X_{\chi'} \to X_{\chi}$ , $\varXi' \to \varXi$ , $\phi_{v'} \colon X_{v'} \to X_v$ , and $\phi_{e'} \colon X_{e'} \to X_e$ such that

fig_28_11

commutes.

Lemma 28: Suppose that every edge map of $\chi'$ is an elevation. Then the map $\Phi$ is $\pi_1$ -injective.

Proof: The idea is to add extra vertex spaces to $\chi'$ so that $\chi'$ satisfies Stalling’s condition. As before, we have inclusions: fig_28_2

If $\chi'$ does not satisfy Stalling’s condition then one of these maps is not surjective. Without loss of generality, suppose that fig_28_31

does not arise as an edge map of $\chi'$ . Suppose $\partial_{e}^{-} \colon X_e \to X_u$ . Then $(\partial_{e}^{-} \circ \phi_{e'})_{*}(\pi_1(X_{e'}))$ is a subgroup of $\pi_1(X_u)$ . Let $X_{u'}$ be the corresponding covering space of $X_u$ . Since $\pi_1(X_{u'})$ contains $(\partial_{e}^{-} \circ \phi_{e'})_{*}(\pi_1(X_{e'}))$ , there is a lift $\partial_{e'} \colon X_{e'} \to X_{u'}$ of $\partial_{e'} \circ \phi_{e'}$ . Now replace $\chi'$ by $\chi' \cup X_{u'}$ and repeat. After infinitely many repetitions, the result $\hat{\chi}$ satisfies the hypothesis of Stalling’s condition, and $\chi'$ is contained in a subgraph of spaces.

Theorem 18: If $G_{+}$ and $G_{-}$ are residually finite groups then $G_{+} * G_{-}$ is also residually finite.

Proof: Let $X_{\pm} = K(G_{\pm}, 1)$ , let $e = *$ be a point, and fix maps $\partial^{\pm} \colon e \to X_{\pm}$ . This defines a graph of spaces $\chi$ . By the Seifert-van Kampen Theorem, $\pi_1(X_{\chi}) \cong G_{+} * G_{-}$ . Let $\tilde{\chi}$ be the graph of spaces structure on the universal cover of $X_{\chi}$ , and let $\Delta \subseteq X_{\tilde{\chi}}$ be a compact subset. We may assume that $\Delta$ is connected; we may also assume that if $\tilde{e} \in \tilde{\varXi}$ (Bass-Serre tree) and $\Delta \cap \tilde{e} \ne \emptyset$ , then $\tilde{e} \subseteq \Delta$ . Let $\eta \colon X_{\tilde{\chi}} \to \tilde{\varXi}$ be the map to the underlying map of the Bass-Serre tree. Then $\eta(\Delta)$ is a finite connected subgraph, $\varXi'$ . For each $v' \subseteq v(\varXi')$ , let $\Delta_{v'} = \Delta \cap X_{v'}$ , a compact subspace of $X_{v'}$ . Because $G_{\pm}$ are residually finite, we have a diagram fig_28_6

where $X_{\hat{v}} \to X_{\pm}$ is a finite-sheeted covering map and $\Delta_{v'}$ embeds in $X_{\hat{v}}$ . Let $\hat{\chi}$ be defined as follows. Set $\hat{\varXi} = \varXi'$ ; for a vertex $\hat{v} \in \hat{\varXi}$ , the vertex space is the $X_{\hat{v}}$ corresponding to $v'$ . For each $\hat{e} \in E(\hat{\varXi})$ corresponding to $e' \in E(\varXi')$ , define $\partial_{\hat{e}}^{\pm}$ so that the diagram fig_28_81

commutes. Now sum.

Exercise 26: If $G_1$ and $G_2$ are residually finite and $H$ is finite, prove that $G_1 *_H G_2$ is residually finite.

27. Stallings’ Condition

6 April 2009 in Course notes | Tags: Elevations, Graphs of groups, Separability | by mathjoker | 1 comment

Recall that our goal is to determine, for a given map of graphs of spaces such as the one shown below, whether the map $\Phi$ can be extended to a covering map $\widehat{\Phi}$ .

Figure 1

Let $\mathcal{X}, \mathcal{X}'$ be graphs of spaces equipped with maps $\Xi' \to \Xi$ , $\phi_{v'} : X_{v'} \to X_v$ and $\phi_{e'} : X_{e'} \to X_e$ as before. Recall that in Stallings’ proof of Hall’s Theorem, we completed an immersion to a covering map by gluing in edges. We will aim to do the same thing with graphs of spaces.

Definition: Let $\mathcal{X}$ be a graph of spaces, and let $\eta : X_{\mathcal{X}} \to \Xi$ be the map to the underlying graph. If $\Delta \subseteq \Xi$ is a subgraph, then $\eta^{-1}(\Delta) \subseteq X_{\mathcal{X}}$ has a graph-of-spaces structure $\mathcal{Y}$ with underlying graph $\Delta$ . Call $\mathcal{Y}$ a subgraph of spaces of $\mathcal{X}$ .

We’re seeking a condition on $\mathcal{X}'$ such that $\mathcal{X}'$ is realized as a subgraph of spaces of some $\widehat{\mathcal{X}}$ with a covering map $\widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}}$ such that the following diagram commutes:

diagram1 Definition: For each edge map $\partial_e^{\pm} : X_e \to X_v$ of $\mathcal{X}$ , and each $v' \mapsto v$ a vertex of $\mathcal{X}'$ , let

$\mathcal{E}^{\pm}(e) = \bigcup_{v' \mapsto v} \mathcal{E}^{\pm}(e, v')$ .

For each possible degree $\mathcal{D}$ , let $\mathcal{E}_{\mathcal{D}}^{\pm}(e) \subseteq \mathcal{E}^{\pm}(e)$ be the set of elevations of degree $\mathcal{D}$ . We will say $\mathcal{X}'$ satisfies Stallings’ condition if and only if the following two things hold:

(a) Every edge map of $\mathcal{X}'$ is an elevation of the appropriate edge map of $\mathcal{X}$ .
(b) For each $e \in E(\Xi)$ and $\mathcal{D}$ , there is a bijection $\mathcal{E}_{\mathcal{D}}^+(e) \leftrightarrow \mathcal{E}_{\mathcal{D}}^-(e)$ .

So in Figure 1, the graph of spaces $\mathcal{X}'$ is something you might be able to turn into a covering. In the picture, $\mathcal{E}_{\mathcal{D}}^+(e)$ is represented by the blue circles, and $\mathcal{E}_{\mathcal{D}}^-(e)$ is represented by the green circles. Observe that the blue circles are in bijection with the green circles.

Corollary: $\mathcal{X}'$ satisfies Stallings’ condition if and only if $\mathcal{X}'$ can be realized as a subgraph of spaces of some $\widehat{\mathcal{X}}$ such that

(a) $V(\Xi ') = V(\widehat{\Xi})$ , and
(b) there is a covering map $\widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}}$ such that the following diagram commutes:

diagram1

Proof of Corollary. First we’ll show that if $\Phi$ can be extended to a covering map as described above, then $\mathcal{X}'$ satisfies Stallings’ condition. By Theorem 17, every edge map of $\widehat{\mathcal{X}}$ is an elevation. So there are inclusions

diagram2

Furthermore, these maps are surjective, and clearly degree-preserving.

Now assume that $\mathcal{X}'$ satisfies Stallings’ condition. Then we build $\widehat{\mathcal{X}}$ as follows. Let $V(\widehat{\Xi}) = V(\Xi ')$ . As above, we have degree-preserving inclusions (this time, not surjections)

diagram3

Extend these inclusions to bijections $\mathcal{E}_{\mathcal{D}}^+(e) \leftrightarrow \mathcal{E}_{\mathcal{D}}^-(e)$ . Now we set $E(\widehat{\Xi}) = \bigcup_{e \in E(\Xi)} \mathcal{E}^+(e)$ . Each of these $\widehat{e}$ is an elevation

diagram4

This defines an edge space $X_{\widehat{e}}$ and an edge map $\partial_{\widehat{e}}^+$ . Consider the corresponding elevation in $\bigcup_{e \in E(\Xi)} \mathcal{E}^-(e)$ :

diagram5

Because $\partial_{\widehat{e}}^+$ and $\partial_{\widehat{e}}^-$ are of the same degree, we have a covering transformation $X_{\widehat{e}} ' \to X_{\widehat{e}}$ . So we can identify them, and use $\partial_e^-$ as the other edge map. By construction, $\widehat{\mathcal{X}}$ satisfies the conditions of Theorem 17, so there is a suitable covering map $\widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}}$ .

Exercise 25: (This will be easier later, but we have the tools necessary to do this now.) Prove that if $G_1$ and $G_2$ are LERF groups, then so is $G_1 * G_2$ .