On the asymptotic equidistribution of word values in symmetric groups

TL;DR

This paper proves that for words with constants in symmetric groups, their values become uniformly distributed as the group size grows, and discusses potential extensions to tuples of elements in free groups.

Contribution

It establishes an asymptotic equidistribution result for word values in symmetric groups, advancing understanding of their distribution properties.

Findings

Word values in symmetric groups become uniformly distributed asymptotically.

Results apply to words with constants in symmetric groups.

Discussion on possible equidistribution for tuples in free groups.

Abstract

We prove an asymptotic equidistribution result for word values for words with constants in the symmetric group. We also speculate about simultaneous asymptotic equidistribution results for values of $d$-tuples of elements of $\mathbb F_d$.