Words, Hausdorff dimension and randomly free groups

Michael Larsen, Aner Shalev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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∈ 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.

Original languageAmerican English
Pages (from-to)1409-1427
Number of pages19
JournalMathematische Annalen
Volume371
Issue number3-4
DOIs
StatePublished - 1 Aug 2018

Bibliographical note

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

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