A note on the structure theory of groups acting on the Bruhat Tits tree associated to $\mathrm{PGL}_{2}(F)$

TL;DR

This paper proves classical decomposition theorems for subgroups of automorphisms of Bruhat-Tits trees related to PGL2 over local fields, aiding the understanding of their representation theory and extending to general Bruhat-Tits trees.

Contribution

It establishes decomposition theorems for subgroups of automorphisms of Bruhat-Tits trees, generalizing to all closed transitive subgroups across any such trees.

Findings

Proved classical decomposition theorems for specific subgroups.

Extended decomposition results to all closed transitive subgroups.

Enhanced understanding of representation theory of these subgroups.

Abstract

In this note we prove the classical decomposition theorems for certain subgroups of the automorphism group of the Bruhat-Tits tree associated to $\mathrm{PGL}_{2}(F)$, where $F$ is a local field. The results can be used to understand the representation theory of these subgroups. Moreover, we can also use the results to obtain similar decompositions for \textit{any} closed transitive subgroup of the automorphism group of \textit{any} Bruhat-Tits tree.