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 $\mathrm {st}_G(n)$. We prove that if $G$ is periodic then the quotients $G/\mathrm {st}_G(n)$ are Beauville groups for every $n\geq 2$ if $p\geq 5$ and $n\geq 3$ 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 $G/\mathrm {st}_G(n)$ are Beauville groups.

中文翻译：

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Grigorchuk–Gupta–Sidki组作为Beauville曲面的来源

如果$ G $是定义在$ p $ -adic树上的Grigorchuk–Gupta–Sidki组，其中$ p $是奇数素数，我们将通过其水平稳定剂研究与$ G $商相关的Beauville曲面的存在$ \ mathrm {st} _G（n）$。我们证明如果$ G $是周期性的，则商$ G / \ mathrm {st} _G（n）$是每个$ n \ geq 2 $的Beauville群，如果$ p \ geq 5 $和$ n \ geq 3 $如果$ p = 3 $。在这种情况下，我们进一步证明，除有限数量外，$ G $的所有商都是Beauville群。另一方面，如果$ G $是非周期性的，则商$ G / \ mathrm {st} _G（n）$都不是Beauville组。