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arXiv · 2107.00589

Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups

Luis Arenas-Carmona·Claudio Bravo·Benoit Loisel·Giancarlo Lucchini Arteche
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Abstract

Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where $\mathbb{F}$ is a finite field, we use a result by Bux, Köhl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications.

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Luis Arenas-Carmona, Claudio Bravo, Benoit Loisel, Giancarlo Lucchini Arteche. 2021-11-23. Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups. https://doi.org/10.1016/j.jpaa.2021.106996

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