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Ping-pong on negatively curved groups. (English) Zbl 0936.20019
In a Kleinian group, any two loxodromic elements with distinct endpoints have high powers which generate a free group on two generators. An algebraic abstraction of this elementary geometric fact gives a sufficient condition for the subgroup generated by two subgroups of a group acting on a set to generate a free product with amalgamation along their intersection. In this paper, the author develops some refinements of this condition in the context of negatively curved groups.
The main result concerns two \(\mu\)-quasiconvex subgroups \(H\) and \(K\) of a \(\delta\)-negatively curved group \(G\). There is a constant \(C\) (depending only on \(G\), \(\delta\), and \(\mu\)) so that for any subgroups \(H_1<H\) and \(K_1<K\) with \(H_1\cap K_1=H\cap K\), if all elements of \(H_1\) and \(K_1\) which are shorter than \(C\) lie in \(H\cap K\), then the subgroup \(\langle H_1,K_1\rangle\) generated by \(H_1\) and \(K_1\) is \(H_1*_{H\cap K}K_1\). In case \(H\) is malnormal (that is, \(gHg^{-1}\cap H=\{1\}\) for any \(g\in G\setminus H\)), and \(K_1=K\), if \(H_1\cap K\) contains all elements of \(H_1\) shorter than a certain constant, then \(\langle H_1,K\rangle\) is \(H_1*_{H\cap K}K\) (that is, there is no requirement on the short elements of \(K\)). In both cases, if \(H_1\) and \(K_1\) are quasiconvex in \(G\), then so is the subgroup they generate.
These results apply to numerous examples. In particular, when \(H\) and \(K\) are residually finite and have trivial intersection, the main result implies that there are infinite families of finite-index subgroups \(H_m\) of \(H\) and \(K_n\) of \(K\) so that \(\langle H_m,K_n\rangle=H_m*K_n\). If \(H\) and \(K\) are LERF, then they need not intersect trivially, and the conclusion becomes that for infinite families of subgroups, \(H_m\cap K_n=H\cap K\) and \(\langle H_m,K_n\rangle=H_m*_{H\cap K}K_n\).

20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
20E26 Residual properties and generalizations; residually finite groups
Full Text: DOI arXiv
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