Teichm¨uller Curves, Triangle Groups, and Lyapunov Exponents
نویسندگان
چکیده
We construct a Teichmüller curve uniformized by the Fuchsian triangle group ∆(m, n, ∞) for every m < n ≤ ∞. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. We interprete some of the so-called Lyapunov exponents of the Kontsevich–Zorich cocycle as normalized degrees of some natural line bundles on a Teichmüller curves. We determine the Lyapunov exponents for the Teichmüller curves we construct.
منابع مشابه
Automorphisms of complexes of curves and of Teichm uller spaces
To every compact orientable surface one can associate following Harvey Ha Ha a combinatorial object the so called complex of curves which is analogous to Tits buildings associated to semisimple Lie groups The basic result of the present paper is an analogue of a fundamental theorem of Tits for these complexes It asserts that every automorphism of the complex of curves of a surface is induced by...
متن کاملA Geometric Approach to the Complex of Curves on a Surface
1. The complex of curves Geometry has played an important role in the study of Teichm uller space and its group of symmetries, the Mapping Class Group of a surface. In 13], Harvey introduced a simplicial complex associated to a surface called the complex of curves, which was intended to encode the asymptotic geometry of Teichm uller space in analogy with similar constructions for symmetric spac...
متن کامل6 Teichmüller Curves , Triangle Groups , and Lyapunov Exponents
We construct a Teichmüller curve uniformized by the Fuchsian triangle group ∆(m, n, ∞) for every m < n ≤ ∞. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find bil-liard tables that generate these Teichmüller curves. We interprete some of ...
متن کاملSchwarz Triangle Mappings and Teichmüller Curves: Abelian Square-tiled Surfaces
We consider normal covers of CP with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian squaretiled surfaces, a...
متن کاملFundamental Domains in Teichm Uller Space
We introduce a natural construction of fundamental domains for actions of subgroups of the mapping class group on Teichm uller space and investigate their properties. These domains are analogous to the classical Dirichlet polyhedra associated to the actions of discrete isometry groups on hyperbolic spaces.
متن کامل