Abstract

We give a method to describe the quotient of the local Bruhat-Tits tree T-P for PGL(2)(K), where K is a global function field, by certain subgroups of PGL(2)(K) of arithmetical significance. In particular, we can compute the quotient of T-P by an arithmetic subgroup PGL(2)(A), where A = A(P) is the ring of functions that are regular outside P, recursively for a place P of any degree, when K is a rational function field. We achieve this by proving that the infinite matrices whose coordinates are the numbers of neighbors of a vertex in T-P corresponding to orders in a fixed isomorphism class commute for different places P, using tools from the theory of representations of orders. The latter result holds for every global function field K. (C) 2013 Elsevier Inc. All rights reserved.