Random extensions of free groups and surface groups are hyperbolic
Random extensions of free groups and surface groups are hyperbolic
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DOI：
10.1093/imrn/rnv138
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发表时间：
2015-01
期刊：
影响因子：
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通讯作者：
Samuel J. Taylor;G. Tiozzo
中科院分区：
文献类型：
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作者：
Samuel J. Taylor;G. Tiozzo
本文证明了秩为N\ge3 $的自由群F_N$或亏格为g\ge2 $的闭可定向曲面S_g$的基本群的随机扩张是双曲群。这里，随机扩张是对应于由$k$独立随机游动生成的Out$（F_N）$或Mod$（S_g）$的子群的扩张。我们的主要定理有几个应用，包括弱双曲群的随机子群是自由的和无畸变的。
In this note, we prove that a random extension of either the free group $F_N$ of rank $N\ge3$ or of the fundamental group of a closed, orientable surface $S_g$ of genus $g\ge2$ 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.