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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 thatfig_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 diagramfig_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 diagramfig_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.

Proof of Theorem 6: Let c \colon [a,b] \to X be a quasi-geodesic in a \delta-hyperbolic space X.  We can replace c by c' \colon [a,b] \to X as in Lemma 8.  Let p = c'(a), q = c'(b), and x \in [p,q] such that D = d(x, \textnormal{im}(c')) is maximal.  We need to bound D.  Let y \in [p,x] be such that d(x,y) = \min \lbrace d(x,p), \: 2D \rbrace, and z \in [x, q] such that d(x,z) = \min \lbrace d(x,q), \: 2D \rbrace.  Let y' \in \textnormal{im}(c') such that d(y,y') \le D, and y' \in \textnormal{im}(c') such that d(z,z') \le D.  Finally, let \gamma be a path in X obtained by concatenating [y,y'], the section of the image of c' with endpoints y' and z', and [z,z'].

proof2ver2

By construction, \gamma does not intersect B(x,D), and part 3 of Lemma 8 shows that

l(\gamma) \le k_1 d(y', z') + k_2 + 2D \le k_1 (6D) + k_2 + 2D.

Lemma 7 gives a bound on the length of paths based on the distance between their endpoints:

D \le \delta \log_2 (l(\gamma)) + 1 \le \delta \log_2 (k_1 (6D) + k_2 + 2D) + 1.

The left hand side of this inequality increases linearly in D, while the right increases logarithmically, so that D must be bounded above for the equality to hold, and clearly this upper bounded depends only on the constants k_1, k_2 and \deltaQED.

It now makes sense to make the following definition.

Definition: A finitely generated group \Gamma is called (word-) hyperbolic if some (any) Cayley graph for \Gamma is Gromov-hyperbolic.  Equivalently, a group \Gamma is hyperbolic if it acts properly discontinuously and cocompactly by isometries on a proper Gromov-hyperbolic metric space.

Examples:

a) Free groups
b) \mathbb{Z}^n is not hyperbolic for n > 1.
c) Let M be any closed hyperbolic manifold.  Then \pi_1(M) is word-hyperbolic.
d)  More generally, \pi_1(M) is word-hyperbolic for any M with negative sectional curvature bounded away from 0.

Without getting into too much detail, we briefly mention the following theorem of Gromov as an indication of how general the class of hyperbolic groups really is.

Theorem (Gromov): A “randomly chosen” finitely presented group is “almost surely” word-hyperbolic.

Definition: A subspace Y of a geodesic metric space X is quasiconvex if there exists a K \ge 0 such that, for all y_1, y_2 \in Y and for all x \in [y_1, y_2], d(x, Y) \le K.

Example: Consider \mathbb{Z}^2 with the \ell^1-metric.  Then the diagonal subgroup \Delta \subset \mathbb{Z}^2 is not quasiconvex (though it is quasi-embedded).

Theorem 6 implies that this kind of poor behavior does not occur in hyperbolic space.

Corollary: Suppose that \Gamma is a word-hyperbolic group and H is a subgroup.  Then H is quasiconvex in some (any) Cayley graph of \Gamma if and only if H is finitely generated and H \hookrightarrow \Gamma is a quasi-embedding.

Proof:  (\Leftarrow) is immediate from Theorem 6.  For the other direction, fix a generating set S for \Gamma, assume H is quasiconvex in the Cayley graph of \Gamma with constant K, and let h be in H.  Consider a geodesic in the Cayley path of \Gamma from 1 to h, which we can take to be of the form s_1...s_n for s_i in S.  Let v_i be the vertices of this geodesic, so v_i= s_1...s_i.

proof3ver2

By quasiconvexity, for each v_i there exists g_i in H such that d(v_i,g_i) \leq K.  Take g_0=1 and g_n=h.  Let h_i=g_{i-1}^{-1}g_i, so h=h_1... h_n.  Let u_i=v_ig_i^{-1}.  Note that l_S(u_i)\leq K.  For each i, we have that h_i=u_is_iu_{i+1}^{-1} and so l_S(h_i)\leq 2K+1.  Therefore, H is generated by T=B(1,2K+1)\cap H, a finite set.  Furthermore, we have shown that l_T(h)\leq n=l_S(h).

But it’s clear that l_S(h)\leq (2K+1)l_T(h) (as each element of T has S-length at most 2K+1) so the inclusion of H into \Gamma is a quasi-isometric embedding.  QED.

In light of this, the following definition makes sense

Definition: A subgroup H of a hyperbolic group \Gamma is called quasiconvex if it is a quasiconvex space of some (any) Cayley graph of \Gamma.

Exercise 14: If G \stackrel{\rho}{\longrightarrow} H is a retraction and G is finitely generated, then the inclusion H \hookrightarrow G is a quasi-isometric embedding.  (Hint: you can choose a generating set S for G such that \rho(s) = s or \rho(s) = 1 for all s \in S.

Example: Marshall Hall’s Theorem implies that every finitely generated subgroup of a free group is a retract of a finite-index subgroup, so every finitely generated subgroup of a free group is quasiconvex.

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