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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 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.

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 Švarc-Milnor 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.


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