Cohomology for groups of isometries of regular trees

In this paper, we study the 1-cohomology groups associated with the unitary irreducible representations of locally compact groups of isometries of regular trees. We begin by explaining definitions and terminology about 1-cohomology groups and Gelfand pairs, already well known in the literature. Next, we focus on the irreducible representations of closed groups of isometries of homogeneous or semihomogeneous trees acting transitively on the tree boundary. We prove that all the groups H1(G,π)H1(G,π) are always zero with only one exception. This result is already known for both groups PGL2(FF) and PSL2(FF) where FF is a local field.