Periodic points of Hamiltonian surface diffeomorphisms

The main result of this paper is that every non-trivial Hamiltonian diffeomorphism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period. The same result is true for S^2 provided the diffeomorphism has at least three fixed points. In addition we show that up to isotopy relative to its fixed point set, every orientation preserving diffeomorphism F: S --> S of a closed orientable surface has a normal form. If the fixed point set is finite this is just the Thurston normal form.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper20.abs.htm

Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups

The main theorem of this document emulates, in the context of Out(F_r) theory, a mapping class group theorem (by H. Masur and J. Smillie) that determines precisely which index lists arise from pseudo-Anosov mapping classes. Since the ideal Whitehead graph gives a finer invariant in the analogous setting of a fully irreducible outer automorphism, we instead focus on determining which of the 21 connected, loop-free, 5-vertex graphs are ideal Whitehead graphs of ageometric, fully irreducible outer automorphisms of the free group of rank 3. Our main theorem accomplishes this by showing that there are precisely 18 graphs arising as such. We also give a method for identifying certain complications called periodic Nielsen paths, prove the existence of conveniently decomposed representatives of ageometric, fully irreducible outer automorphisms having connected, loop-free, (2r-1)-vertex ideal Whitehead graphs, and prove a criterion for identifying representatives of ageometric, fully irreducible outer automorphisms. The methods we use for constructing fully irreducible outer automorphisms of free groups, as well as our identification and decomposition techniques, can be used to extend our main theorem, as they are valid in any rank. Our methods of proof rely primarily on Bestvina-Feighn-Handel train track theory and the theory of attracting laminations

On train track splitting sequences

We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This result is an important step in the proof that the disk complex is Gromov hyperbolic. As another application we show that train track sliding and splitting sequences give quasi-geodesics in the train track graph, generalizing a result of Hamenstaedt [Invent. Math.].Comment: 40 pages, 12 figure

Uniform Hyperbolicity of the Graphs of Curves

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some $\delta=\delta(g,p)$. In this paper, we show that there exists some $\delta>0$ independent of $g,p$ such that the curve graph $\mathcal{C}_{1}(S_{g,p})$ is $\delta$-hyperbolic. Furthermore, we use the main tool in the proof of this theorem to show uniform boundedness of two other quantities which a priori grow with $g$ and $p$: the curve complex distance between two vertex cycles of the same train track, and the Lipschitz constants of the map from Teichm\"{u}ller space to $\mathcal{C}(S)$ sending a Riemann surface to the curve(s) of shortest extremal length.Comment: 19 pages, 2 figures. This is a second version, revised to fix minor typos and to make the end of the main proof more understandabl