Intrinsic Rank in CAT(0) Spaces

TL;DR

This paper establishes a strong notion of rank in proper, geodesically complete CAT(0) spaces satisfying duality, showing that most geodesics have parallel sets that are flat Euclidean spaces of a fixed dimension.

Contribution

It introduces a unique rank for such CAT(0) spaces by demonstrating that generic parallel sets are isometric to Euclidean spaces of a specific dimension.

Findings

Most geodesics have parallel sets isometric to Euclidean space

A unique dimension k characterizes the flatness of parallel sets

Euclidean space embeds in all parallel sets

Abstract

Let $X$ be a proper, geodesically complete CAT(0) space which satisfies Chen and Eberlein's duality condition. We show the existence of a strong notion of rank for $X$ by proving that the parallel sets $P_v$ of geodesics $v$ in $X$ are generically flat. More precisely, let $GX$ be the space of parametrized unit-speed geodesics in $X$. There is a unique $k$ and a dense $G_\delta$ set $\mathcal{A}$ in $GX$ such that $P_v$ is isometric to flat Euclidean space $\mathbb{R}^k$, for all $v \in \mathcal{A}$. It follows that $\mathbb{R}^k$ isometrically embeds in $P_v$ for every $v \in GX$.