Grigorchuk-Gupta-Sidki groups as a source for Beauville surfaces

TL;DR

This paper investigates when quotients of Grigorchuk-Gupta-Sidki groups over p-adic trees form Beauville surfaces, establishing conditions based on the group's periodicity and the prime p.

Contribution

It characterizes the conditions under which these quotients are Beauville groups, linking group periodicity and prime p to the existence of Beauville surfaces.

Findings

Quotients are Beauville groups for periodic G when p≥5 and n≥2, or p=3 and n≥3.

Non-periodic G quotients are never Beauville groups.

Provides criteria connecting group properties to geometric structures.

Abstract

If $G$ is a Grigorchuk-Gupta-Sidki group defined over a $p$-adic tree, where $p$ is an odd prime, we study the existence of Beauville surfaces associated to the quotients of $G$ by its level stabilizers $\st_G(n)$. We prove that if $G$ is periodic then the quotients $G/\st_G(n)$ are Beauville groups for every $n\geq 2$ if $p\geq 5$ and $n\geq 3$ if $p=3$. On the other hand, if $G$ is non-periodic, then none of the quotients $G/\st_G(n)$ are Beauville groups.