The Frequency Space of a Free Group

We analyze the structure of the frequency spaceQ(F) of a nonabelian free group F = F(a1,…,ak) consisting of all shift-invariant Borel probability measures on ∂F and construct a natural action of Out(F) on Q(F). In particular we prove that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ‖ϕ(w)‖/‖w‖, where w is a nontrivial conjugacy class, is the intersection of ℚ and a closed subinterval of ℝ with rational endpoints.

[1]  Rita Gitik,et al.  On the Combination Theorem for Negatively Curved Groups , 1996, Int. J. Algebra Comput..

[2]  M. Burger,et al.  Rigidity in Dynamics and Geometry , 2002 .

[3]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[4]  A. Furman.  Coarse-Geometric Perspective on Negatively Curved Manifolds and Groups , 2002 .

[5]  Michael Handel,et al.  Laminations, trees, and irreducible automorphisms of free groups , 1997 .

[6]  F. Bonahon.  The geometry of Teichmüller space via geodesic currents , 1988 .

[7]  Hyperbolic automorphisms of free groups , 1999, math/9906008.

[8]  Francis Bonahon.  Bouts des Varietes Hyperboliques de Dimension 3 , 1986 .

[9]  Robert H. Gilman,et al.  Geometric and Computational Perspectives on Infinite Groups , 1995 .

[10]  R. Lyons.  Equivalence of boundary measures on covering trees of finite graphs , 1994, Ergodic Theory and Dynamical Systems.

[11]  M. Bestvina,et al.  The topology at infinity of Out(Fn) , 2000 .

[12]  K. Vogtmann,et al.  Moduli of graphs and automorphisms of free groups , 1986 .

[13]  Vincent Guirardel.  Dynamics of Out(Fn) on the boundary of outer space , 2000 .

[14]  K. R. Parthasarathy,et al.  On the category of ergodic measures , 1961 .

[15]  F. Bonahon.  Geodesic currents on negatively curved groups , 1991 .

[16]  Ilya Kapovich,et al.  Generic properties of Whitehead’s algorithm and isomorphism rigidity of random one-relator groups , 2003 .

[17]  Mladen Bestvina,et al.  Train tracks and automorphisms of free groups , 1992 .

[18]  Boris Hasselblatt,et al.  Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION: WHAT IS LOW-DIMENSIONAL DYNAMICS? , 1995 .

[19]  M. Lustig,et al.  IRREDUCIBLE AUTOMORPHISMS OF $F_{n}$ HAVE NORTH–SOUTH DYNAMICS ON COMPACTIFIED OUTER SPACE , 2003, Journal of the Institute of Mathematics of Jussieu.

[20]  J. Zukas.  Introduction to the Modern Theory of Dynamical Systems , 1998 .

[21]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[22]  Reiner Martin.  Non-uniquely ergodic foliations of thin type , 1997, Ergodic Theory and Dynamical Systems.

[23]  Mladen Bestvina,et al.  Addendum and correction to: ``A combination theorem for negatively curved groups'' , 1996 .

[24]  Mladen Bestvina,et al.  A combination theorem for negatively curved groups , 1992 .

[25]  M. Bridson,et al.  The symmetries of outer space , 2001 .

[26]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.