ℝ‐trees and laminations for free groups I: algebraic laminations

This paper is the first of a sequence of three papers, where the concept of a real tree dual to a measured geodesic lamination in a hyperbolic surface is generalized to arbitrary real trees provided with a (very small) action of a free group by isometries. Laminations for free groups are defined with care in three different approaches: algebraic laminations, symbolic laminations, and laminary languages. The topology on the space of laminations and the action of the outer automorphisms group are detailed

[1]  Michael Handel,et al.  The Tits alternative for Out(Fn) II: A Kolchin type theorem , 1997 .

[2]  Philippe Narbel.  The boundary of Iterated Morphisms on Free Semi-Groups , 1996, Int. J. Algebra Comput..

[3]  Pavel Zorin-Kranich,et al.  Habilitationsschrift , 1970 .

[4]  Michael Handel,et al.  The Tits alternative for Out (F~n) I: Dynamics of exponentially-growing automorphisms , 1997 .

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

[6]  M. Lustig,et al.  $\R$-trees and laminations for free groups III: Currents and dual $\R$-tree metrics , 2008 .

[7]  C. Mauduit,et al.  Substitutions in dynamics, arithmetics, and combinatorics , 2002 .

[8]  W. Lickorish.  AUTOMORPHISMS OF SURFACES AFTER NIELSEN AND THURSTON (London Mathematical Society Student Texts 9) , 1990 .

[9]  Gilbert Levitt,et al.  An index for counting fixed points of automorphisms of free groups , 1998 .

[10]  Ilya Kapovich.  The Frequency Space of a Free Group , 2005, Int. J. Algebra Comput..

[11]  A. Fathi,et al.  Travaux de Thurston sur les surfaces : séminaire Orsay , 1991 .

[12]  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.

[13]  M. Lustig,et al.  ℝ‐trees and laminations for free groups III: currents and dual ℝ‐tree metrics , 2007, 0706.0677.

[14]  F. Beaufils,et al.  FRANCE , 1979, The Lancet.

[15]  Pierre Arnoux,et al.  Fractal representation of the attractive lamination of an automorphism of the free group , 2006 .

[16]  É. Ghys,et al.  Sur Les Groupes Hyperboliques D'Apres Mikhael Gromov , 1990 .

[17]  Ilya Kapovich.  Currents on free groups , 2004, math/0412128.

[18]  Daryl Cooper,et al.  Automorphisms of free groups have finitely generated fixed point sets , 1987 .