Words, Hausdorff dimension and randomly free groups

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 $1 \ne w \in F_d$ there exists $\epsilon > 0$ such that if $\Gamma$ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $w:\Gamma ^d \rightarrow \Gamma$ have Hausdorff dimension at most $d -\epsilon$. We conclude that profinite groups $G := {\hat{\Gamma }}$, $\Gamma$ as above, satisfy no probabilistic identity, and therefore they are randomly free, namely, for any $d \ge 1$, the probability that randomly chosen elements $g_1, \ldots , g_d \in G$ 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.