Morse boundaries of proper geodesic metric spaces

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 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 Teichmüller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichmüller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichmüller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichmüller space into the Thurston compactication of Teichmüller space by projective measured foliations.