Strongly Contracting Geodesics in Outer Space

Abstract

We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out(F_n) act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a quasi-geodesic with endpoints on the axis stays within a bounded distance from the axis.

37 pages. Revised applications chapter

Cited by in corpus (19)

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• Hyperbolic extensions of free groups
• Lines of minima in Outer space
• Topology and Dynamics of the Contracting Boundary of Cocompact CAT(0) Spaces
• The isometry group of Outer Space
• Growth Tight Actions
• A Metrizable Topology on the Contracting Boundary of a Group
• The Poisson boundary of $\text{Out}(F_N)$
• Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups
• Central limit theorems for mapping class groups and $\text{Out}(F_N)$
• Sublinearly Morse Geodesics in CAT(0) Spaces: Lower Divergence and Hyperplane Characterization
• Spectral theorems for random walks on mapping class groups and $\text{Out}(F_N)$
• On the acylindrical hyperbolicity of the tame automorphism group of ${\mathrm{SL}}_2(\mathbb{C})$
• Random trees in the boundary of Outer space
• Sublinearly Morse Boundary II: Proper geodesic spaces
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Counting conjugacy classes of fully irreducibles: double exponential growth
• Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space
• Quasi-geodesics in Out(F_n) and their shadows in sub-factors