# Norm growth for the Busemann cocycle

## TL;DR

This paper provides an explicit upper bound for the norm of the Busemann cocycle on regular trees, revealing its asymptotic linearity in relation to the square root of vertex distance, with connections to harmonic cocycles.

## Contribution

It introduces a new explicit upper bound for the Busemann cocycle norm on trees, highlighting symmetries and asymptotic behavior, and relates to harmonic cocycles.

## Key findings

- The Busemann cocycle norm grows asymptotically linearly with the square root of vertex distance.
- The cocycle takes values in a submodule related to the Steinberg representation.
- An explicit upper bound for the cocycle norm is established.

## Abstract

Using explicit methods, we provide an upper bound to the norm of the Busemann cocycle of a locally finite regular tree $X$, emphasizing the symmetries of the cocycle. The latter takes value into a submodule of square summable functions on the edges of $X$, which corresponds the Steinberg representation for rank one groups acting on their Bruhat-Tits tree. The norm of the Busemann cocycle is asymptotically linear with respect to square root of the distance between any two vertices. Independently, Gournay and Jolissaint proved an exact formula for harmonic 1-cocycles covering the present case.