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Quasiconvexity and amalgams. (English) Zbl 0912.20031
Let \(G\) be a word hyperbolic group equipped with a finite symmetric set of generators \(S\), with corresponding Cayley graph \(\Gamma\) equipped with its simplicial metric. Let \(A\) be a subgroup of \(G\). Then \(A\) is said to be quasiconvex in \(G\) if there exists \(\varepsilon\geq 0\) such that for all \(a\in A\), for every geodesic path in \(\Gamma\) joining the identity to \(a\) and for all \(x\) on this geodesic path, we can find \(a'\in A\) such that \(d(x,a')\leq\varepsilon\). It is well-known that this property of \(A\) is independent of the choice of the generating set \(S\).
The author proves the following Theorem 1. Suppose that the hyperbolic group \(G\) is an amalgamated product, \(G=A_{-1}*_C A_1\), with \(C\) being a cyclic subgroup, and let \(H\) be a finitely generated subgroup of \(G\). Then, the following are equivalent: (a) \(H\) is quasiconvex in \(G\); (b) for all \(g\in G\) and for all \(i=1,-1\), the subgroup \(gHg^{-1}\cap A_i\) is quasiconvex in \(A_i\). (As the author notes, \(A_1\) and \(A_{-1}\) are quasiconvex subgroups in \(G\).)
The author proves then several consequences of Theorem 1, concerning in particular the so-called exponential groups which have been studied by A. Myasnikov and V. Remeslennikov. Other important consequences concern the groups which have property \((Q)\). We recall that a hyperbolic group \(G\) is said to have property \((Q)\) if any finitely generated subgroup of \(G\) is quasiconvex in \(G\). The author shows how to obtain new groups with property \((Q)\) from existing ones.
He proves the following result using Theorem 1: Theorem 2. Suppose that \(G=A_{-1}*_C A_1\) is word hyperbolic, with \(C\) virtually cyclic and \(A_{-1}\) and \(A_1\) having property \((Q)\). Then, \(G\) has property \((Q)\). The techniques used involve Bass-Serre trees, graphs of groups and normal forms.

20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
57M07 Topological methods in group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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