The metric completion of Outer Space

We develop the theory of a metric completion of an asymmetric metric space. We characterize the points on the boundary of Outer Space that are in the metric completion of Outer Space with the Lipschitz metric. We prove that the simplicial completion, the subset of the completion consisting of simpli...

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Bibliographic Details
Published in:Geometriae dedicata 2020-02, Vol.204 (1), p.191-230
Main Author: Algom-Kfir, Yael
Format: Article
Language:English
Subjects:
Algebraic Geometry
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Metric space
Original Paper
Projective Geometry
Topology
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Summary:We develop the theory of a metric completion of an asymmetric metric space. We characterize the points on the boundary of Outer Space that are in the metric completion of Outer Space with the Lipschitz metric. We prove that the simplicial completion, the subset of the completion consisting of simplicial tree actions, is homeomorphic to the free splitting complex. We use this to give a new proof of a theorem by Francaviglia and Martino that the isometry group of Outer Space is isomorphic to Out ( F n ) for n ≥ 3 and to PSL ( 2 , Z ) for n = 2 .
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-019-00451-3