48 research outputs found
Return words of linear involutions and fundamental groups
We investigate the natural codings of linear involutions. We deduce from the
geometric representation of linear involutions as Poincar\'e maps of measured
foliations a suitable definition of return words which yields that the set of
first return words to a given word is a symmetric basis of the free group on
the underlying alphabet . The set of first return words with respect to a
subgroup of finite index of the free group on is also proved to be a
symmetric basis of
Geodesic laminations revisited
The Bratteli diagram is an infinite graph which reflects the structure of
projections in a C*-algebra. We prove that every strictly ergodic unimodular
Bratteli diagram of rank 2g+m-1 gives rise to a minimal geodesic lamination
with the m-component principal region on a surface of genus g greater or equal
to 1. The proof is based on the Morse theory of the recurrent geodesics on the
hyperbolic surfaces.Comment: 13 pages, 2 figures, revised versio
Fractal representation of the attractive lamination of an automorphism of the free group
N°RR 05066 (2005)International audienceIn this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers ({\it iwip}) automorphisms) is given in the case where the dilation coefficient of the automorphism is a unit Pisot number. The shift map associated with the attractive symbolic lamination is, in this case, proved to be measure-theoretically isomorphic to a domain exchange on a self-similar Euclidean compact set. This set is called the central tile of the automorphism, and is inspired by Rauzy fractals associated with Pisot primitive substitutions. The central tile admits some specific symmetries, and is conjectured under the Pisot hypothesis to be a fundamental domain for a toral translation
Infinite sequence of fixed point free pseudo-Anosov homeomorphisms
We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed
points and leaving invariant a sequence of orientable measured foliations on
the same topological surface and the same stratum of the space of abelian
differentials. The existence of such sequences show that all pseudo-Anosov
homeomorphisms fixing orientable measured foliations cannot be obtained by the
Rauzy-Veech induction strategy
A random tunnel number one 3-manifold does not fiber over the circle
We address the question: how common is it for a 3-manifold to fiber over the
circle? One motivation for considering this is to give insight into the fairly
inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds
with tunnel number one, we provide compelling theoretical and experimental
evidence that fibering is a very rare property. Indeed, in various precise
senses it happens with probability 0. Our main theorem is that this is true for
a measured lamination model of random tunnel number one 3-manifolds.
The first ingredient is an algorithm of K Brown which can decide if a given
tunnel number one 3-manifold fibers over the circle. Following the lead of
Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by
working in the context of train tracks/interval exchanges. To analyze the
resulting algorithm, we generalize work of Kerckhoff to understand the dynamics
of splitting sequences of complete genus 2 interval exchanges. Combining all of
this with a "magic splitting sequence" and work of Mirzakhani proves the main
theorem.
The 3-manifold situation contrasts markedly with random 2-generator 1-relator
groups; in particular, we show that such groups "fiber" with probability
strictly between 0 and 1.Comment: This is the version published by Geometry & Topology on 15 December
200