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 $\e > 0$ 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 $d -\e$.
We conclude that profinite groups $G := \hat\Gamma$, $\Gamma$ as above, satisfy no probabilistic identity, and therefore they are \emph{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 \cite{DPSS}.
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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On representations of Gal(\overline{Q}/Q), \hat{GT} and Aut(\hat{F}_2).
By work of Belyi, the absolute Galois group G_Q of the rational numbers embeds into a subgroup \hat{GT} called the Grothendeick-Teichmuller group of the group A of continuous automorphisms of a profinite group on two generators. We show that a rich class of representations of G_Q lifts to \hat{GT} by showing they lift all the way to a finite index subgroup of A.
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- Award ID(s):
- 1701785
- NSF-PAR ID:
- 10356543
- Date Published:
- Journal Name:
- Journal of algebra
- Volume:
- 607
- Issue:
- A
- ISSN:
- 1090-266X
- Page Range / eLocation ID:
- 134-159
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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