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Property \(P_{\mathrm{naive}}\) for acylindrically hyperbolic groups. (English) Zbl 1515.20238

Summary: We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the \(P_{\mathrm{naive}}\) property: for any finite collection of elements \(h_1, \ldots , h_k\), there exists another element \(\gamma \ne 1\) such that for all \(i\), \(\langle h_i, \gamma \rangle = \langle h_i \rangle * \langle \gamma \rangle\). We also show that if a collection of subgroups \(H_1, \dots , H_k\) is a hyperbolically embedded collection, then there is \(\gamma \ne 1\) such that for all \(i\), \(\langle H_i, \gamma \rangle = H_i * \langle \gamma \rangle \).

MSC:

20F67 Hyperbolic groups and nonpositively curved groups
20E07 Subgroup theorems; subgroup growth
20F65 Geometric group theory
46L35 Classifications of \(C^*\)-algebras
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