Beauville structures in $p$-central quotients

TL;DR

This paper proves that certain $p$-central quotients of free groups and free products are Beauville groups for primes $p \\geq 5$, and identifies Beauville structures for $p=3$, expanding known examples.

Contribution

It confirms Boston's conjecture for $p \\geq 5$ and explicitly constructs new Beauville $3$-groups, providing new examples in the field.

Findings

All $p$-central quotients of specified groups are Beauville groups for $p \\geq 5$

Explicit Beauville structures identified for $p=3$

Provides an infinite family of Beauville $3$-groups

Abstract

We prove a conjecture of Boston that if $p\geq 5$, all $p$-central quotients of the free group on two generators and of the free product of two cyclic groups of order $p$ are Beauville groups. In the case of the free product, we also determine Beauville structures in $p$-central quotients when $p=3$. As a consequence, we give an explicit infinite family of Beauville $3$-groups, which is different from the only one that was known up to date.