Divergence of Morse geodesics

TL;DR

This paper constructs specific CAT(0) spaces with Morse geodesics exhibiting divergence functions of any power greater than or equal to 2, answering a question about the possible divergence rates of Morse geodesics.

Contribution

It demonstrates the existence of CAT(0) spaces with Morse geodesics having divergence functions of any power s ≥ 2, filling a gap in understanding divergence behaviors.

Findings

Existence of Morse geodesics with divergence equivalent to r^s for any s ≥ 2

Construction of CAT(0) spaces with prescribed divergence rates

Answers a previously open question about divergence functions in CAT(0) spaces

Abstract

Behrstock and Dru\c{t}u raised a question about the existence of Morse geodesics in $CAT(0)$ spaces with divergence function strictly greater than $r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than $1$. In this paper, we answer the question of Behrstock and Dru\c{t}u by showing that for each real number $s\geq 2$, there is a $CAT(0)$ space $X$ with a proper and cocompact action of some finitely generated group such that $X$ contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.