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Francaviglia, Stefano; Martino, Armando

Metric properties of outer space. (English) Zbl 1268.20042

Publ. Mat., Barc. 55, No. 2, 433-473 (2011).
Summary: We define metrics on Culler-Vogtmann space, which are an analogue of the Thurston metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices.
We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms.
We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer Space, quasi-geodesic for the symmetric metric.

Cited in 2 Reviews
Cited in 37 Documents

MSC:

20F65 Geometric group theory
57M07 Topological methods in group theory
20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups

Keywords:

Culler-Vogtmann space; outer space; free groups; Thurston metric; Lipschitz metric; stretching factors; optimal maps; marked metric graphs; automorphisms; geodesics
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\textit{S. Francaviglia} and \textit{A. Martino}, Publ. Mat., Barc. 55, No. 2, 433--473 (2011; Zbl 1268.20042)
Full Text: DOI arXiv Euclid
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