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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.

##### MSC:
 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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