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Preprints, Working Papers, ... Year : 2021

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

Luis Arenas-Carmona
  • Function : Author
Claudio Bravo
  • Function : Author
Benoit Loisel
Giancarlo Lucchini Arteche
  • Function : Author

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, K\"ohl 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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Dates and versions

hal-03277266 , version 1 (02-07-2021)

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Luis Arenas-Carmona, Claudio Bravo, Benoit Loisel, Giancarlo Lucchini Arteche. Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups. 2021. ⟨hal-03277266⟩
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