Quasiflats in CAT(0) complexes

TL;DR

This paper investigates the geometric structure of quasiflats in certain Euclidean complexes, revealing their asymptotic conical nature and local flatness outside compact regions, under symmetry conditions.

Contribution

It establishes that 2-quasiflats in cocompact piecewise Euclidean 2-complexes are close to asymptotically conical, locally flat subsets, advancing understanding of their large-scale geometry.

Findings

2-quasiflats are at finite Hausdorff distance from asymptotically conical sets

Such quasiflats are locally flat outside compact sets

Results apply to complexes with cocompact isometry groups

Abstract

We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.