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

Luis Arenas-Carmona , Claudio Bravo , Benoit Loisel (1) , Giancarlo Lucchini Arteche

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Mots clés: special unitary groups; quotient graphs; Bruhat-Tits trees; arithmetic subgroups; algebraic function fields

Domaines: Théorie des groupes [math.GR]; Théorie des nombres [math.NT]

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⟩