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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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