Hecke algebras from groups acting on trees and HNN extensions

TL;DR

This paper investigates the structure of Hecke algebras arising from groups acting on trees, providing explicit algebraic descriptions and exploring their C*-completions, with implications for totally disconnected locally compact groups.

Contribution

It offers a unified geometric framework for understanding Hecke algebras associated with groups acting on trees, including explicit multiplication tables and C*-algebra properties.

Findings

Explicit multiplication tables for Hecke algebra generators

Conditions for Hecke algebra to have a universal C*-completion

Unified geometric approach connecting algebraic and analytic methods

Abstract

We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal C*-completion. The paper unifies past algebraic and analytic approaches by focusing on the common geometric thread.The results have implications for the general theory of totally disconnected locally compact groups.