Teichmuller curves, triangle groups, and Lyapunov exponents

• 成果类型: Article
• 署名作者: Bouw, Irene I.; Moeller, Martin
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.172.139
• 发表日期: 2010
• 页码: 139-185
• 关键词: fuchsian-groups; ergodic averages; billiards; SURFACES; SPACE; VARIETIES; covers; FLOWS

摘要

We construct a Teichmuller curve uniformized by a Fuchsian triangle group commensurable to Delta(m, n, infinity) for every m, n <= infinity. In most cases, for example when m not equal n and m or n is odd, the uniformizing group is equal to the triangle group Delta(m, n, infinity). Our construction includes the Teichmuller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find billiard tables that generate these Teichmuller curves. We interpret some of the so-called Lyapunov exponents of the Kontsevich-Zorich cocycle as normalized degrees of a natural line bundle on a Teichmuller curve. We determine the Lyapunov exponents for the Teichmuller curves we construct.