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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| Published in | Inventiones mathematicae Vol. 228; no. 3; pp. 1309 - 1351 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.06.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | .
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
. |
|---|---|
| AbstractList | In this paper we introduce a new height on P1(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 ωω+1. . 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 . |
| ArticleNumber | 1309 |
| Author | McMullen, Curtis T. |
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| Cites_doi | 10.1007/BF02992821 10.4064/aa-56-2-93-110 10.1353/ajm.2007.0002 10.1007/BF02992820 10.3934/jmd.2008.2.209 10.1515/crelle-2021-0019 10.1017/etds.2018.55 10.1007/BF01344143 10.1093/qjmath/53.1.75 10.4310/jdg/1299766791 10.3934/jmd.2009.3.611 10.3934/jmd.2013.7.1 10.1215/S0012-7094-06-13335-5 10.1007/978-1-4757-1810-2 10.1090/jams/950 10.1215/S0012-7094-06-13326-4 10.4007/annals.2010.172.139 10.1007/BF01388890 10.4171/055-1/11 10.1515/crelle-2019-0037 10.1007/978-3-319-04075-2 10.1007/978-3-540-31347-2_13 10.1090/S0894-0347-05-00512-6 10.1007/BF02941497 10.1090/S0894-0347-03-00432-6 10.1007/978-1-4612-1210-2 10.4007/annals.2018.188.1.5 10.1017/S0143385700004521 10.1007/978-1-4757-6720-9 10.1007/BF02392964 10.1007/BF01220377 10.1007/BF01565437 |
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| Snippet | .
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... In this paper we introduce a new height on P1(K) associated to an Abelian variety with real multiplication by K, and use it to study non-arithmetic triangle... |
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| SubjectTerms | Arithmetic Billiards Mathematics Mathematics and Statistics Multiplication Semigroups |
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| Title | Billiards, heights, and the arithmetic of non-arithmetic groups |
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