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As before, are and are graphs of spaces equipped with maps , , , and such that
Lemma 28: Suppose that every edge map of is an elevation. Then the map is -injective.
Proof: The idea is to add extra vertex spaces to so that satisfies Stalling’s condition. As before, we have inclusions:
If does not satisfy Stalling’s condition then one of these maps is not surjective. Without loss of generality, suppose that
does not arise as an edge map of . Suppose . Then is a subgroup of . Let be the corresponding covering space of . Since contains , there is a lift of . Now replace by and repeat. After infinitely many repetitions, the result satisfies the hypothesis of Stalling’s condition, and is contained in a subgraph of spaces.
Theorem 18: If and are residually finite groups then is also residually finite.
Proof: Let , let be a point, and fix maps . This defines a graph of spaces . By the Seifert-van Kampen Theorem, . Let be the graph of spaces structure on the universal cover of , and let be a compact subset. We may assume that is connected; we may also assume that if (Bass-Serre tree) and , then . Let be the map to the underlying map of the Bass-Serre tree. Then is a finite connected subgraph, . For each , let , a compact subspace of . Because are residually finite, we have a diagram
where is a finite-sheeted covering map and embeds in . Let be defined as follows. Set ; for a vertex , the vertex space is the corresponding to . For each corresponding to , define so that the diagram
commutes. Now sum.
Exercise 26: If and are residually finite and is finite, prove that is residually finite.
Proof of Theorem 6: Let be a quasi-geodesic in a -hyperbolic space . We can replace by as in Lemma 8. Let , , and such that is maximal. We need to bound . Let be such that , and such that . Let such that , and such that . Finally, let be a path in obtained by concatenating , the section of the image of with endpoints and , and .
By construction, does not intersect , and part 3 of Lemma 8 shows that
Lemma 7 gives a bound on the length of paths based on the distance between their endpoints:
The left hand side of this inequality increases linearly in , while the right increases logarithmically, so that must be bounded above for the equality to hold, and clearly this upper bounded depends only on the constants , and . QED.
It now makes sense to make the following definition.
Definition: A finitely generated group is called (word-) hyperbolic if some (any) Cayley graph for is Gromov-hyperbolic. Equivalently, a group is hyperbolic if it acts properly discontinuously and cocompactly by isometries on a proper Gromov-hyperbolic metric space.
a) Free groups
b) is not hyperbolic for .
c) Let be any closed hyperbolic manifold. Then is word-hyperbolic.
d) More generally, is word-hyperbolic for any with negative sectional curvature bounded away from .
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 of a geodesic metric space is quasiconvex if there exists a such that, for all and for all , .
Example: Consider with the -metric. Then the diagonal subgroup 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 is a word-hyperbolic group and is a subgroup. Then is quasiconvex in some (any) Cayley graph of if and only if is finitely generated and is a quasi-embedding.
Proof: is immediate from Theorem 6. For the other direction, fix a generating set for , assume is quasiconvex in the Cayley graph of with constant , and let be in . Consider a geodesic in the Cayley path of from to , which we can take to be of the form for in . Let be the vertices of this geodesic, so .
By quasiconvexity, for each there exists in such that . Take and . Let , so . Let . Note that . For each , we have that and so . Therefore, is generated by , a finite set. Furthermore, we have shown that .
But it’s clear that (as each element of has -length at most ) so the inclusion of into is a quasi-isometric embedding. QED.
In light of this, the following definition makes sense
Definition: A subgroup of a hyperbolic group is called quasiconvex if it is a quasiconvex space of some (any) Cayley graph of .
Exercise 14: If is a retraction and is finitely generated, then the inclusion is a quasi-isometric embedding. (Hint: you can choose a generating set for such that or for all .
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.