Hadamard spaces with isolated flats

TL;DR

This paper studies nonpositively curved spaces with isolated flats, showing their boundary invariance, group properties like relative hyperbolicity and biautomaticity, and characterizing such spaces via their Tits boundary.

Contribution

It characterizes spaces with isolated flats as relatively hyperbolic and relates their boundary structure to geometric properties, extending results to more general settings.

Findings

The geometric boundary is an invariant of the group.

Groups acting on these spaces are relatively hyperbolic and biautomatic.

A CAT(0) space has isolated flats iff its Tits boundary is a union of points and spheres.

Abstract

We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that any such group is relatively hyperbolic, biautomatic, and satisfies the Tits Alternative. The main step in establishing these results is a characterization of spaces with isolated flats as relatively hyperbolic with respect to flats. Finally we show that a CAT(0) space has isolated flats if and only if its Tits boundary is a disjoint union of isolated points and standard Euclidean spheres. In an appendix written jointly with Hindawi, we extend many of the results of this article to a more general setting in which the isolated subspaces are not required to be flats.