A note on the structure theory of groups acting on the Bruhat Tits tree associated to ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(F)$

In this note we prove the classical decomposition theorems for certain subgroups of the automorphism group of the Bruhat-Tits tree associated to ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(F)$, 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 $\mathit{\text{any}}$ closed transitive subgroup of the automorphism group of $\mathit{\text{any}}$ Bruhat-Tits tree.