# Typical behaviour along geodesic rays in hyperbolic groups

## TL;DR

This paper investigates the asymptotic behavior of real-valued functions on hyperbolic groups along typical geodesic rays, with applications to homomorphisms, quasimorphisms, and convex cocompact actions.

## Contribution

It introduces new insights into the limiting behavior of functions on hyperbolic groups along geodesic rays, extending understanding to various types of functions and actions.

## Key findings

- Characterization of limiting behavior of functions along geodesic rays
- Application to group homomorphisms and quasimorphisms
- Analysis of displacement functions in convex cocompact actions

## Abstract

In this note we study the limiting behaviour of real valued functions on hyperbolic groups as we travel along typical geodesic rays in the Gromov boundary of the group. Our results apply to group homomorphisms, certain quasimorphisms and to the displacement functions associated to convex cocompact group actions on CAT$(-1)$ metric spaces.