Words, Hausdorff dimension and randomly free groups
Words, Hausdorff dimension and randomly free groups
复制标题
词、Hausdorff 维数和随机自由群
DOI：
10.1007/s00208-017-1635-y
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发表时间：
2018
影响因子：
1.4
通讯作者：
Shalev, Aner
中科院分区：
文献类型：
--
作者：
Larsen, Michael;Shalev, Aner
We bound the size of fibers of word maps in finite and residually finite groups, and derive various applications. Our main result shows that, for any wordthere existssuch that ifis a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word maphave Hausdorff dimension at most. We conclude that profinite groups,as above, satisfy no probabilistic identity, and therefore they arerandomly free, namely, for any, the probability that randomly chosen elementsfreely generate a free subgroup (isomorphic to) is 1. This solves an open problem from Dixon et al. (J Reine Angew Math (Crelle’s) 556:159–172, 2003). Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit–Thompson theorem.
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DOI：
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发表时间：
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期刊：
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DOI：
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