Morse Boundaries of Proper Geodesic Metric Spaces

TL;DR

This paper introduces the Morse boundary, a new hyperbolic boundary concept for proper geodesic spaces, and explores its properties and applications to Teichmüller space and mapping class groups, revealing complex topological structures.

Contribution

It defines the Morse boundary as a quasi-isometry invariant and connects it to known boundaries, providing new insights into Teichmüller space and mapping class groups.

Findings

Morse boundary is homeomorphic for mapping class group and Teichmüller space.

Morse boundaries of Teichmüller space can contain high-dimensional spheres.

There exists an injective continuous map into the Thurston compactification.

Abstract

We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper $\mathrm{CAT}(0)$ space this boundary is the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichm\"uller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichm\"uller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichm\"uller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichm\"uller space into the Thurston compactification of Teichm\"uller space by projective measured foliations. An appendix includes a corrigendum to the paper introducing refined Morse gauges to correct the proof and statement of Lemma 2.10.