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 word 1 ≠ w∈ Fd there exists ϵ> 0 such that if Γ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map w: Γ d→ Γ have Hausdorff dimension at most d- ϵ. We conclude that profinite groups G: = Γ ^ , Γ as above, satisfy no probabilistic identity, and therefore they are randomly free, namely, for any d≥ 1 , the probability that randomly chosen elements g1, … , gd∈ G freely generate a free subgroup (isomorphic to Fd) 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.
Bibliographical noteFunding Information:
ML was partially supported by NSF Grant DMS-1401419. AS was partially supported by ERC advanced Grant 247034, BSF Grant 2008194, ISF Grant 1117/13 and the Vinik Chair of mathematics which he holds.
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