Morse Quasiflats I

TL;DR

This paper introduces Morse quasiflats, generalizing Morse quasigeodesics to higher dimensions, establishing their equivalence under certain conditions, and providing foundational examples for future geometric analysis.

Contribution

It defines multiple notions of Morse quasiflats, proves their equivalence and invariance under quasi-isometries, and offers foundational examples in the context of geometric group theory.

Findings

Definitions of Morse quasiflats are equivalent under certain conditions

Morse quasiflats are quasi-isometry invariant

Examples illustrating Morse quasiflats are provided

Abstract

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.