Random Extensions of Free Groups and Surface Groups Are Hyperbolic
نویسنده
چکیده
In this note, we prove that a random extension of either the free group FN of rank N ě 3 or of the fundamental group of a closed, orientable surface Sg of genus g ě 2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either OutpFN q or ModpSgq generated by k independent random walks. Our main theorem is that a k–generated random subgroup of ModpSgq or OutpFN q 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.
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