Word Measures and Profinite Rigidity

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• Liam Hanany, University of Cambridge
• Friday 17 October 2025, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Will Cohen.

Given a word $w \in F_{r}$ in a free group on $r$-generators, and a finite group $G$, the word map $G^{r} \to G$ is the map obtained by evaluating the word $w$ on the $r$-tuple of elements of $G$. The word map yields a measure on the finite group $G$ by pushing forward the uniform measure on $G^{r}$. We will discuss these measures and their connection to conjectures regarding profinite rigidity of words in free groups. We will then describe some techniques used to prove special cases of these conjectures by studying word measures on symmetric groups. If time permits, we will also discuss how these ideas are related to solutions of equations in free groups and the non-negative immersions of graphs of free groups with cyclic edge groups.

This talk is part of the Junior Geometry Seminar series.

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