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Free subgroups of groups with nontrivial Floyd boundary. (English) Zbl 1036.20032
Let $$\Gamma$$ be a finitely generated group, let $$S$$ be a finite generating set for $$\Gamma$$ and let $$C(\Gamma,S)$$ be the associated Cayley graph equipped with its simplicial metric $$d$$ which induces the word metric on $$\Gamma$$. A ‘Floyd metric’ on $$C(\Gamma,S)$$ is obtained by scaling the metric $$d$$ in the following manner: we choose a vertex $$v$$ in $$C(\Gamma,S)$$ and we multiply the length of each edge $$e$$ by a factor which is a function of $$r=r(v,e)$$ = the least number of edges between $$v$$ and $$e$$, which satisfies $$\lambda f(r)\leq f(r+1)\leq f(r)$$ for some $$\lambda>0$$ and for any integer $$r$$ and $$\sum_{r=0}^\infty f(r)<\infty$$. (For instance, we can take the function $$f(r)=r^{-2}$$.) The resulting metric on $$C(\Gamma,S)$$ is not complete and the Floyd metric $$d'$$ is the length metric on $$C(\Gamma,S)$$ associated to the new set of lengths of edges. The metric completion $$\overline{C(\Gamma,S)}$$ is the ‘Floyd completion’ of the group. The set $$\overline{C(\Gamma,S)}\setminus C(\Gamma,S)$$ is ‘Floyd’s boundary’ $$\partial\Gamma$$. The definition and the basic properties are contained in [W. J. Floyd, Invent. Math. 57, 205-218 (1980; Zbl 0428.20022)]. A subgroup $$\Lambda$$ of $$\Gamma$$ is said to be ‘elementary with respect to $$\partial\Gamma$$’ if there exists a sequence $$g_n$$ in $$\Lambda$$ such that both $$g_n$$ and $$g_n^{-1}$$ converge to points of $$\partial\Gamma$$ and $$\Lambda$$ does not fix (setwise) this set of limit points.
In the paper under review, the author, generalizing Floyd’s construction, defines Floyd’s boundary for a group generated by a set which could be countable. With the appropriate definitions, he proves the following:
Theorem 1. Let $$\Gamma$$ be a group generated by a finite or countable set $$S$$ and let $$\Lambda$$ be a subgroup. Assume that $$\Lambda$$ is nonelementary with respect to $$\partial\Gamma$$ and that every infinite subset of $$\Lambda$$ is unbounded with respect to the word metric. Then $$\Lambda$$ contains a noncommutative free subgroup.
Theorem 2. Let $$\Gamma$$ be a finitely generated group, let $$S$$ be a finite generating set of $$\Gamma$$ and assume that $$\partial\Gamma$$ contains at least three points. Then $$\partial\Gamma$$ is a boundary of $$\Gamma$$ in the sense of Furstenberg and $$\Gamma$$ acts on $$\partial\Gamma$$ as a convergence group.
The author notes that some of the ideas and results in the paper are contained in M. Gromov’s paper [in Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)].

##### MSC:
 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 20E07 Subgroup theorems; subgroup growth 20F05 Generators, relations, and presentations of groups
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