Random extensions of free groups and surface groups are hyperbolic

2015/01/12 by Taylor, Samuel J., Tiozzo, Giulio
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1501.02846

Abstract

In this note, we prove that a random extension of either the free group F_N of rank N≥3 or of the fundamental group of a closed, orientable surface S_g of genus g≥2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either Out(F_N) or Mod(S_g) generated by k independent random walks. Our main theorem has several applications, including that a random subgroup of a weakly hyperbolic group is free and undistorted.

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