# zbMATH — the first resource for mathematics

Metric properties of outer space. (English) Zbl 1268.20042
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.

##### MSC:
 20F65 Geometric group theory 57M07 Topological methods in group theory 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups
Full Text:
##### References:
 [1] Y. Algom-Kfir, Strongly contracting geodesics in Outer Space (2008), arXiv: · Zbl 1250.20019 [2] M. Bestvina and M. Feighn, Outer limits, Preprint (1994). [3] M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7(2) (1997), 215\Ndash244. · Zbl 0884.57002 [4] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135(1) (1992), 1\Ndash51. · Zbl 0757.57004 [5] M. Bestvina and K. Fujiwara, A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19(1) (2009), 11\Ndash40. · Zbl 1203.53041 [6] M. M. Cohen and M. Lustig, Very small group actions on $$\mathbf{R}$$-trees and Dehn twist automorphisms, Topology 34(3) (1995), 575\Ndash617. · Zbl 0844.20018 [7] D. Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111(2) (1987), 453\Ndash456. · Zbl 0628.20029 [8] M. Culler and J. W. Morgan, Group actions on $$\mathbf{R}$$-trees, Proc. London Math. Soc. (3) 55(3) (1987), 571\Ndash604. · Zbl 0658.20021 [9] S. Francaviglia, Geodesic currents and length compactness for automorphisms of free groups, Trans. Amer. Math. Soc. 361(1) (2009), 161\Ndash176. · Zbl 1166.20032 [10] V. Guirardel and G. Levitt, Deformation spaces of trees, Groups Geom. Dyn. 1(2) (2007), 135\Ndash181. · Zbl 1134.20026 [11] I. Kapovich, The frequency space of a free group, Internat. J. Algebra Comput. 15(5-6) (2005), 939\Ndash969. · Zbl 1110.20031 [12] I. Kapovich, Currents on free groups, in: “Topological and asymptotic aspects of group theory” , Contemp. Math. 394 , Amer. Math. Soc., Providence, RI, 2006, pp. 149\Ndash176. · Zbl 1110.20034 [13] K. Vogtmann, Automorphisms of free groups and Outer Space, in: “Proceedings of the Conference on Geometric and Combinatorial Group Theory” , Part I (Haifa, 2000), Geom. Dedicata 94 (2002), 1\Ndash31. · Zbl 1017.20035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.