Lifts of simple curves in finite regular coverings of closed surfaces

Veech surfaces and simple closed curves

We study the SL(2,R)–infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the “no small virtual triangles” theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense i...

Coverings of singular curves over finite fields

Pseudo-anosov Maps and Simple Closed Curves on Surfaces

Suppose C and C are two sets of simple closed curves on a hyperbolic surface F . We will give necessary and sufficient conditions for the existence of a pseudo-Anosov map g such that g(C) ∼= C. Pseudo-Anosov maps are the most important class among surface homeomorphisms [Th3], and they play important roles in 3-manifold theory [Th2]. This note is to address the following question: Suppose F is ...

Group Actions, Coverings and Lifts of Automorphisms Group Actions, Coverings and Lifts of Automorphisms

a Supported in part by \Ministrstvo za znanost in tehnologijo Slovenije", proj.no. Abstract The problem of lifting automorphisms of a (fairly arbitrary) topological space to automorphisms of a covering space is studied by means of voltage groups. To this end we introduce some required background results regarding group actions and covering spaces.

Vertices of Closed Curves in Riemannian Surfaces

We study the relation between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M . In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms...