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

We show the mapping class group, CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic grou...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2022-02, Vol.300 (2), p.1557-1602
Main Authors: Russell, Jacob, Spriano, Davide, Tran, Hung Cong
Format: Article
Language:English
Subjects:
Geodesy
Mapping
Mathematics
Mathematics and Statistics
Subgroups
Theorems
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Summary:We show the mapping class group, CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-021-02811-w