Morse Boundaries of Proper Geodesic Metric Spaces
By Matthew Cordes. 2015
94 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/366dd7db59802c71ff16f443a835ac0cf2f27c20
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In Computational Theory and Mathematics
- Dominik Gruber
- Christopher H. Cashen
- Goulnara Arzhantseva
- David Hume
- Carolyn R. Abbott
- Jason A. Behrstock
- Hung Cong Tran
- Denis V. Osin
In Geometry and Topology
- Hung Cong Tran
- Carolyn R. Abbott
- Dominik Gruber
- Goulnara Arzhantseva
- Christopher H. Cashen
- David Hume
- Jason A. Behrstock
- Wen-yuan Yang
Citations in the corpus
- On strongly quasiconvex subgroups (2017, background): “TRONGLY QUASICONVEX SUBGROUPS 4 In order to prove analogies of the above properties for strongly quasiconvex subgroups, we need a quasi-isometry invariant boundary for any finitely generated group. In [Cor17], Cordes introduced the Morse boundary of proper geodesic metric spaces which generalizes the”
- Characterizations of Morse Quasi-Geodesics via Superlinear Divergence and Sublinear Contraction (2016, methodology): “tion has been a topic of intense interest in relation to mapping class groups and outer automorphisms of free groups [1, 6], acylindrically hyperbolic groups [13,25], and contracting/Morse boundaries [27,28,11,12,22]. We introduce a more general version of contracting projection than has been previo”
- Largest acylindrical actions and Stability in hierarchically hyperbolic groups (2017, methodology): “Importantly, it is a quasi-isometry invariant and generalizes the Gromov boundary of a hyperbolic space [Cor15].”