Functions conditionally of negative type on groups acting on regular trees

TL;DR

This paper investigates functions conditionally of negative type on groups acting on regular trees, providing optimal bounds for cocycle growth and characterizing unbounded functions of this type.

Contribution

It introduces an optimal upper bound for cocycle growth in groups acting on regular trees and characterizes unbounded functions conditionally of negative type.

Findings

Established an optimal growth bound for cocycles in these groups.

Provided a description of unbounded functions conditionally of negative type.

Connected cocycle bounds with projections of the Haagerup cocycle.

Abstract

Let $\mathcal{T}_{q+1}$ be the $(q+1)$-regular tree and let $G$ be a group of automorphisms acting transitively on the vertices and on the boundary of $\mathcal{T}_{q+1}$. We give an upper bound for the growth of cocycles with values in any unitary representation of the group $G$. This bound is optimal by projecting the Haagerup cocycle onto an appropriate subspace of $\ell^{2}(E)$. We also obtain a description of functions conditionally of negative type which are unbounded.