# Grigorchuk–Gupta–Sidki groups as a source for Beauville surfaces

### Şükran Gül

TED University, Çankaya/Ankara, Turkey### Jone Uria-Albizuri

BCAM - Basque Center for Applied Mathematics, Bilbao, Spain

## 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 $n≥2$ if $p≥5$ and $n≥3$ if $p=3$. In this case, we further show that all but finitely many quotients of $G$ are Beauville groups. On the other hand, if $G$ is non-periodic, then none of the quotients $G/st_{G}(n)$ are Beauville groups.

## Cite this article

Şükran Gül, Jone Uria-Albizuri, Grigorchuk–Gupta–Sidki groups as a source for Beauville surfaces. Groups Geom. Dyn. 14 (2020), no. 2, pp. 689–704

DOI 10.4171/GGD/559