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Lemma 29: Suppose $\{f_i':C_i\longrightarrow\Gamma^{H}\}$ is a finite set of infinite degree elvations and $\Delta \subseteq \Gamma^{H}$ is compact. Then for all sufficiently large $d>0$, there exists an intermediate covering $\Gamma_d$ such that

(a) $\Delta$ embeds in $\Gamma_d$

(b) every $f'_i$ descends to an elevation $\hat{f_i}:\hat{C_i}\longrightarrow \Gamma_d$ of degree $d$

(c) the $\hat{f_i}$ are pairwise distinct

Proof: We claim that the images of $f_i'$ never share an infinite ray (a ray is an isometric embedding of $[0,\infty)$). Neither do two ends of the same elevation $f_i'$. Let’s claim by passing to the universal cover of $\Gamma$, a tree $T$.

For each $i$, lift $f_i'$ to a map $\tilde{f_i}:\mathbb{R}\longrightarrow T$. If $f'_i$ and $f'_j$ share an infinite ray then there exists $h\in H$ such that $\tilde{f_i}$ and $h\tilde{f_j}$ overlay in an infinite ray. The point is that $\tilde{f_i}, \tilde{f_j}$ correspond to cosets $g_if_{\ast}(\pi_1(C))$ and $g_jf_{\ast}(\pi_1(C))$. But this implies that

$g_if_{\ast}(\pi_1(C))=hg_jf_{\ast}(\pi_1(C))$

This implies that $Hg_if_{\ast}(\pi_1(C))=Hg_jf_{\ast}(\pi_1(C))$. So $f'_i=f'_j$. A similar argument implies that the two ends of $f'_i$ do not overlap in an infinite ray. This proves the claim.

Let $\Gamma'$ be the core of $\Gamma^{H}$. Enlarging $\Delta$ if necessary, we can assume that

(i) $\Gamma'\subseteq\Delta$;

(ii) $\Delta$ is a connected subgraph;

(iii) for each $i$, for some $x_i\in C'_i$, $f'_i(x_i)\in\Delta$;

(iv) for each $i$, $|im(f'_i)\cap \delta\Delta|=2$.

For each $i$ identifying $C'_i$ with $\mathbb{R}$ so that $C$ is identified with $\mathbb{R}/\mathbb{Z}$ and $x_i$ is identified with $0$. Let

$\Delta_d=\Delta\cup(\bigcup_i f'_i([-d/2,d/2]))$

For all sufficiently large $d$,

$f'_i(\pm d/2)\notin\Delta$

Now, the restriction of $\Delta_d \longrightarrow \Delta$ factors through $\Delta_d/\sim\longrightarrow\Gamma$, where $f'_i(d/2)\sim f'_i(-d/2)$. This is a finite-to-one immersion, so, by theorem 5, we can complete it to a finite-sheeted covering map as required. $\Box$

Theorem 19: $D$ is LERF.

Recall the set-up from the previous lecture. We built a graph of spaces $\mathscr{X}$ for $D$.

Proof: Let $H\subset D$ be finitely generated. Let $X_H$ be the corresponding covering space of $X_{\mathscr{X}}$ and let $\Delta\subseteq X_H$ be compact. Because $H$ is finitely generated, there exists a subgraph of spaces $X'$ such that $\pi_1(X') =H$. We can take $X'$ large enough so that $\Delta \subseteq X'$. We can enlarge $\Delta$ so that it contains every finite-degree edge space of $X'$. Also enlarge $\Delta$ so that

$\partial^{\pm}_{e'}(\Delta\times (\mathrm{interval}))\subseteq\Delta$

for any $e'\in E(\Xi')$. For each $v'\in V(\Xi')$ let $\Delta_{v'}=\Delta\cap X_{v'}$ and let ${f'_i}=\{$ incident edge map of infinite degree $\}$.

Applying lemma 29 to $\Gamma^H=X_{v'}$, for some large $d$, set $X_{\hat{v}}=\Gamma_d$. (Here we use the fact that vertex groups of $\mathscr{X}'$ are finitely generated)

Define $\mathscr{X}^+$ as follows:

$\bullet \Xi^+=\Xi^-$

$\bullet$ For each $v^+\in V(\Xi^+)$, the edge space is the $X_{v^+}$ that the lemma produced from the corresponding $v'$.

Now, by construction, $\bigcup_{v^+}X_{v^+}$ can be completed to a graph of spaces $\mathscr{X}^+$ so that the map

$X_{\mathscr{X}'}\longrightarrow X_{\mathscr{X}}$

factors through $X_{\mathscr{X}'}\longrightarrow X_{\mathscr{X}}$ and $\Delta$ embeds. Let $\mathscr{X}^-$ be identical to $\mathscr{X}^+$ except with +’s and -‘s exchanged. Clearly $\mathscr{X}^+\cup\mathscr{X}^-$ satisfies Stallings condition, as required. $\Box$

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 $f$:

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

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.

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:

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:

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

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)

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

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

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 $G_1 * G_2$.