### The local-to-global property for Morse quasi-geodesics.

Algebra and Geometry Seminar

4th December 2019, 2:30 pm – 3:30 pm

Fry Building, 2.04

An important property of Gromov hyperbolic spaces is the fact that every path whose all sufficiently long subpaths are quasi-geodesics, needs to be a quasi-geodesic itself. Gromov showed that this property is actually a characterization of hyperbolic spaces. In this talk, we will consider a weakened version of this local-to-global behavioru, called the Morse local-to-global property. The class of spaces that satisfy the Morse local-to-global property include several examples of interest, such as CAT(0) spaces, Mapping Class Groups, fundamental groups of closed 3-manifolds and so on. The leverage offered by knowing that a space satisfies this property allows to import several results and techniques from the theory hyperbolic groups. In particular, we obtain results on different aspects such as stable subgroups, normal subgroups and algorithmic properties.

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