Random extensions of free groups and surface groups are hyperbolic

In this note, we prove that a random extension of either the free group FN of rank N ≥ 3 or of the fundamental group of a closed, orientable surface Sg of genus g≥2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either Out(FN) or Mod(Sg) generated by k independent random walks. Our main theorem is that a k-generated random subgroup of Mod(Sg) or Out(FN) 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.