Random extensions of free groups and surface groups are hyperbolic

الأصل · EN

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 has several applications, including that a random subgroup of a weakly hyperbolic group is free and undistorted.