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Random extensions of free groups and surface groups are hyperbolic
Taylor, SJ Tiozzo, G
Taylor, SJ
Tiozzo, G
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Journal Article
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2016-01-01
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10.1093/imrn/rnv138
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© The Author(s) 2015. Published by Oxford University Press. 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.
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International Mathematics Research Notices
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