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Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2
Author(s) -
Maclachlan C.
Publication year - 2011
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdq085
Subject(s) - mathematics , fuchsian group , dimension (graph theory) , arithmetic , reflection (computer programming) , discrete group , group (periodic table) , genus , reflection group , pure mathematics , hyperbolic space , field (mathematics) , algebraic number field , arithmetic progression , relatively hyperbolic group , hyperbolic manifold , algebra over a field , discrete mathematics , hyperbolic function , mathematical analysis , coxeter group , chemistry , botany , organic chemistry , coxeter element , computer science , biology , programming language
Here we show that if Γ is an arithmetic Fuchsian group of genus 0, then the totally real defining field k of Γ must be such that [ k : ℚ] ⩽ 11. The same inequality holds for discrete arithmetic hyperbolic reflection groups acting on a two‐dimensional hyperbolic space ℍ 2 . In addition, we show that there exists an arithmetic Fuchsian group of genus 0 containing an element of order N if and only if N ∈ {2, 3, …, 16, 18, 20, 22, 24, 26, 28, 30, 36}. A slightly less precise statement holds for discrete arithmetic hyperbolic reflection groups acting on ℍ 2 .
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