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Random extensions of free groups and surface groups are hyperbolic. (English) Zbl 1368.20060
Summary: In this note, we prove that a random extension of either the free group \(F_N\) of rank \(N\geq 3\) or of the fundamental group of a closed, orientable surface \(S_g\) of genus \(g\geq 2\) is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either \(\mathrm{Out}(F_N)\) or \(\mathrm{Mod}(S_g)\) generated by \(k\) independent random walks. Our main theorem is that a \(k\)-generated random subgroup of \(\mathrm{Mod}(S_g)\) or \(\mathrm{Out}(F_N)\) is free of rank \(k\) and convex cocompact. More generally, we show that a \(k\)-generated random subgroup of a weakly hyperbolic group is free and undistorted.

20F67 Hyperbolic groups and nonpositively curved groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20E05 Free nonabelian groups
20E22 Extensions, wreath products, and other compositions of groups
20P05 Probabilistic methods in group theory
20E07 Subgroup theorems; subgroup growth
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