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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 $$\langle H_m,K_n\rangle=H_m*_{H\cap K}K_n$$.

##### MSC:
 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
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