Billiards, heights, and the arithmetic of non-arithmetic groups

. In this paper we introduce a new height on P 1 ( K ) associated to an Abelian variety with real multiplication by K , and use it to study non-arithmetic triangle groups, Teichmüller curves, and billiards in lattice polygons. Complementary results on matrix coefficients and measures are obtained us...

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Bibliographic Details
Published inInventiones mathematicae Vol. 228; no. 3; pp. 1309 - 1351
Main Author McMullen, Curtis T.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2022
Springer Nature B.V
Subjects
Arithmetic
Billiards
Mathematics
Mathematics and Statistics
Multiplication
Semigroups
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Summary:. In this paper we introduce a new height on P 1 ( K ) associated to an Abelian variety with real multiplication by K , and use it to study non-arithmetic triangle groups, Teichmüller curves, and billiards in lattice polygons. Complementary results on matrix coefficients and measures are obtained using modular symbols. In particular, we show the matrix entries m of the classical Hecke group Δ ( 2 , 5 , ∞ ) are constrained by the condition that - γ - 2 ( m ′ / m ) lies in a countable, closed semigroup S ⊂ [ - 1 , 1 ] homeomorphic to ω ω + 1 .
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-022-01101-4