Cyclic hyperbolic Veech groups in finite area

We prove that there are finite area flat surfaces whose Veech group is an infinite cyclic group consisting of hyperbolic elementsComment: 7 pages, 2 figure

Length of a curve is quasi-convex along a Teichmuller geodesic

We show that for every simple closed curve \alpha, the extremal length and the hyperbolic length of \alpha are quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.Comment: 25 pages, 2 figure

The shadow of a Thurston geodesic to the curve graph

We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curves along the two geodesics are not identical.Comment: 34 pages, 10 figures, minor revisions, final version appears in Journal of Topolog

Hyperbolicity in Teichm\"uller space

We review and organize some results describing the behavior of a Teichm\"uller geodesic and draw several applications: 1) We show that Teichm\"uller geodesics do not back track. 2) We show that a Teichm\"uller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpoints are not necessarily in the thick part. 3) We show that if an edge of a Teichm\"uller geodesic triangle passes through the thick part, then it is close to one of the other egdes.Comment: 26 pages, 2 figure

New solutions to the Hurwitz problem on square identities

The Hurwitz problem of composition of quadratic forms, or of "sum of squares identity" is tackled with the help of a particular class of $(\mathbb{Z}_2)^n$-graded non-associative algebras generalizing the octonions. This method provides an explicit formula for the classical Hurwitz-Radon identity and leads to new solutions in a neighborhood of the Hurwitz-Radon identity.Comment: 13 pages, 2 figures, final version to appear in J. Pure Appl. Al

Teichmuller geodesics that do not have a limit in PMF

We construct a Teichmuller geodesic which does not have a limit on the Thurston boundary of the Teichmuller space.Comment: published versio