## Abstract

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∈ F_{d} 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 g_{1}, … , g_{d}∈ G freely generate a free subgroup (isomorphic to F_{d}) 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.

Original language | American English |
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Pages (from-to) | 1409-1427 |

Number of pages | 19 |

Journal | Mathematische Annalen |

Volume | 371 |

Issue number | 3-4 |

DOIs | |

State | Published - 1 Aug 2018 |

### Bibliographical note

Funding 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.

Publisher Copyright:

© 2017, Springer-Verlag GmbH Germany, part of Springer Nature.