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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.

We will see two examples of non-quasiconvex subgroups in this section. The first one is NOT a hyperbolic group, while the second one is.

Example: For the first example, let

$A=\left(\begin{array}{cc}2&1\\1&1\end{array}\right)$,

with one eigenvalue (the larger one) $\lambda>1$. Notice that $A$ does not fix any non-zero vectors in $\mathbf{Z}^2$ (such a map $A$ is called Anosov).

Now let $\Gamma_A=\mathbf{Z}^2\rtimes_A\mathbf{Z}=(\langle a\rangle\oplus\langle b\rangle)\rtimes_A\langle t\rangle$. This is a group.  The group law works like this: for any $g\in\mathbf{Z}^2$, $tgt^{-1}=Ag$. Pick $S=\{a,b\}$, $T=\{a,b,t\}$. The map $\mathbf{Z}^2\hookrightarrow\Gamma_A$ is, by the following analysis,  NOT a quasi-embedding:

Choose $g\in\mathbf{Z}^2$ such that $\lim_{n\rightarrow\infty}\displaystyle\frac{||A^ng||_2}{\lambda^n||g||_2}=1$. All norms on $\mathbf{R}^2$ are bilipschitz, so there exists $k\geq 1$ such that $k^{-1}||g||_2\leq||g||_1\leq k||g||_2$. Therefore, for sufficiently large $n$$||A^ng||_1\geq k^{-1}||A^ng||_2\geq k^{-2}\lambda^n||g||_2\geq k^{-3}\lambda^n||g||_1$, and so $l_S(A^ng)\geq k^{-3}\lambda^n l_S(g)$. On the other side, we have $l_T(t^ngt^{-n})\leq l_T(g)+2n$. It follows that $\mathbf{Z}^2\hookrightarrow\Gamma_A$ is not a quasi-embedding.

Example: For the second example, let $\Sigma$ be a hyperbolic surface. An automorphism $\psi$ of $\Sigma$ is called pseudo-Anosov if for any smooth closed curve $\gamma$ on $\Sigma$ and any $n\in \mathbf{Z}\smallsetminus\{0\}$, $\psi^n(\gamma)$ is not homotopic to $\gamma$. Let $M_{\psi}$ be the mapping torus of $\psi$, i.e., $M_{\psi}:=\Gamma\times [0,1]/\sim$, with the relation $\sim$ generated by $(x,0)\sim (\psi(x),1)$.

Under these assumptions, we are able to use a theorem of Thurston asserting that, $M$ must be a hyperbolic 3-manifold. (W. Thurston, “On the geometry and dynamics of diffeomorphisms of surfaces,” Bull. Amer. Math. Soc. vol 19 (1988), 417-431)

Hence, if $\Gamma$ is closed, then $M_{\psi}$ is also closed. So $\pi_1(M_{\psi})$ acts nicely on $\mathbf{H}^3$ (actually $\pi_1(M_{\psi})=\pi_1(\Sigma)\rtimes_{\psi_*}\mathbf{Z}$), and so is word-hyperbolic by the ŠvarcMilnor Lemma. Then a similar argument to the previous shows the natural map $\pi_1(\Sigma)\hookrightarrow\pi_1(M_{\psi})$ is NOT a quasi-embedding.

For concrete examples, see A. Casson & S. Bleiler, “Automorphisms of Surfaces After Nielsen and Thurston”.

After the two examples, let us switch to a property for all hyperbolic groups:

Theorem 7: Hyperbolic groups are finitely presented.

In order to prove this theorem, we need the following lemma:

Lemma 9: Let $c,c': [0,T]\rightarrow X$ be two geodesics in a $\delta$-hyperbolic metric space $X$, $c(0)=c'(0)$.  (If $c$ is longer than $c'$, say, then extend $c'$ by the constant map). Then for any $t\in [0,T]$, $d(c(t),c'(t))\leq 2(\delta+d(c(T),c'(T)))$.

Proof: Case 1: there is $t'\in [0,T]$ such that $d(c(t),c'(t'))\leq\delta$. Without loss of generality, assume $t'>t$, then $|t'-t|=d(c'(t'),c'(0))-d(c'(t),c'(0))\leq$ $d(c'(t'),c(t))+d(c(t),c(0))-d(c'(t),c'(0))=d(c'(t'),c(t))+t-t=$ $d(c'(t'),c(t))\leq\delta$. So, $d(c(t),c'(t))\leq d(c(t),c'(t'))+d(c'(t'),c'(t))\leq$ $\delta+|t-t'|\leq 2\delta$.

Case 2: there is no $t'\in [0,T]$ such that $d(c(t),c'(t'))\leq\delta$. Then $c(t)$ must be within distance $\delta$ of $[c(T),c'(T)]$. Apply a similar argument to the previous, we see $d(c(t),c'(t))\leq 2(\delta+d(c(T),c'(T)))$.

Proof of Theorem 7: Let $\Gamma$ be $\delta$-hyperbolic, with the generating set $S$. Let $w$ be any relation, which corresponds to a loop in the Cayley graph $Cay_s(\Gamma)$. We can always take $w=\gamma s\delta^{-1}$ with $\gamma$ and $\delta$ geodesics in $Cay_s(\Gamma)$ and $s\in S$, by “triangulating”.

Write $\gamma_t=\gamma(t)$$\delta_t=\delta(t)$. Denote $u_t=\gamma_t\cdot\gamma_{t-1}^{-1}$$v_t=\delta_t\cdot\delta_{t-1}^{-1}$$\alpha_t=\delta_t\cdot\gamma_t^{-1}$.  An easy induction shows that

$w=\prod_{t=1}^T\delta_{t-1}\alpha_{t-1}^{-1}u_t\alpha_t v_t^{-1}\delta_{t-1}^{-1}$.

But Lemma 9 implies that $l(\alpha_t)\leq 2(\delta+1)$ for all $t$, so we have written the loop $w$ as a product of conjugates of loops of length at most $4\delta+6$.  Therefore, the set of all loops of length at most $4\delta+6$ is a finite set of relations for $\Gamma$.