14 research outputs found
Strongly Contracting Geodesics in Outer Space
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.Comment: 37 pages. Revised applications chapte
Mapping tori of small dilatation irreducible train-track maps
We prove that for every P there is a bound B depending only on P so that the
mapping torus of every P--small irreducible train-track map can be obtained by
surgery from one of B mapping tori. We show that given an integer P>0 there is
a bound depending only on P, so that there exists a presentation of the
fundamental group of the mapping torus of a P--small irreducible train-track
map with less than M generators and M relations.Comment: Some figures in colo
The visual boundary of hyperbolic free-by-cyclic groups
Let be an atoroidal outer automorphism of the free group . We
study the Gromov boundary of the hyperbolic group . We explicitly describe a family of embeddings of
the complete bipartite graph into . To do so, we
define the directional Whitehead graph and prove that an indecomposable
-tree is Levitt type if and only if one of its directional Whitehead
graphs contains more than one edge. As an application, we obtain a direct proof
of Kapovich-Kleiner's theorem that is homeomorphic to the
Menger curve if the automorphism is atoroidal and fully irreducible.Comment: 25 pages, 3 figure
Digraphs and cycle polynomials for free-by-cyclic groups
Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be
represented by an expanding, irreducible train-track map. The automorphism
determines a free-by-cyclic group
and a homomorphism . By work of Neumann,
Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, has an open cone
neighborhood in whose integral points
correspond to other fibrations of whose associated outer automorphisms
are themselves representable by expanding irreducible train-track maps. In this
paper, we define an analog of McMullen's Teichm\"uller polynomial that computes
the dilatations of all outer automorphism in .Comment: 41 pages, 20 figure