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15 March 2008 A geometric characterization of arithmetic Fuchsian groups
Slavyana Geninska, Enrico Leuzinger
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Duke Math. J. 142(1): 111-125 (15 March 2008). DOI: 10.1215/00127094-2008-002

Abstract

The trace set of a Fuchsian group Γ encodes the set of lengths of closed geodesics in the surface Γ\H. Luo and Sarnak [3] showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering (BC) property. Sarnak [5] then conjectured that the BC property actually characterizes arithmetic Fuchsian groups. Schmutz [6] stated the even stronger conjecture that a cofinite Fuchsian group is arithmetic if its trace set has linear growth. He proposed a proof of this conjecture in the case when the group Γ contains at least one parabolic element, but unfortunately, this proof contains a gap. In this article, we point out this gap, and we prove Sarnak's conjecture under the assumption that the Fuchsian group Γ contains parabolic elements.

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Slavyana Geninska. Enrico Leuzinger. "A geometric characterization of arithmetic Fuchsian groups." Duke Math. J. 142 (1) 111 - 125, 15 March 2008. https://doi.org/10.1215/00127094-2008-002

Information

Published: 15 March 2008
First available in Project Euclid: 27 March 2008

zbMATH: 1141.20029
MathSciNet: MR2397884
Digital Object Identifier: 10.1215/00127094-2008-002

Subjects:
Primary: 11F06 , 20H10
Secondary: 22E40 , 30F35

Rights: Copyright © 2008 Duke University Press

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Vol.142 • No. 1 • 15 March 2008
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