Abstract

Let K be the function field of a curve C over a field F of either odd or zero characteristic. Following the work by Serre and Mason on SL2, we study the action of arithmetic subgroups of 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 F is a finite field, we use a result by Bux, Kohl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications. (C) 2021 Elsevier B.V. All rights reserved.