Arithmetic groups and the length spectrum of Riemann surfaces
type: Item
instance of: scholarly article
title: Arithmetic groups and the length spectrum of Riemann surfaces
zbMATH Open document ID: 0867.30030
DOI: 10.1215/S0012-7094-96-08407-0
published in: Duke Mathematical Journal
publication date: 1997-08-04T00:00:00Z
review text: This crucial paper should be read by anyone with even a passing interest in arithmetic groups. The author, who now also goes under the names ``Paul Schmutz Schaller and Paul S. Schaller'', offers here a striking result achieved by novel techniques, as well as a summary important by previous results. The theorem is: Let \(M=H/ \Gamma\). Define \(Tr (\Gamma) =Tr(M): =\{2 \cosh (L(a)/2)\): \(a\) a closed geodesic of \(M\} \cup\{2\}\), where \(L(a)\) stands for the length of \(a\). Theorem. Let \(\Gamma\) be a cofinite Fuchsian group which contains at least one parabolic element. Then (i) \(\Gamma\) is an arithmetic group if and only if there exists a finite constant \(C\) such that \(\#\{a\in Tr(\Gamma): a\leq n\} \leq 1+Cu\), \(\forall n\geq 0\). (ii) \(\Gamma\) is an arithmetic group derived from a quaternion algebra if and only if \(\text{Gap} (\Gamma): =\inf\{|a-b |:a,b\in Tr(\Gamma)\), \(a\neq b\} >0\). To give a bit of background, it has been known since Huber (see, for example, Peter Buser, Geometry and spectra of compact Riemann surfaces (1992; Zbl 0770.53001), see Chapter 9) that the number of geodesics with trace \(\leq T\) is \(T^2/2 \log(T)+ O(T^{2-\alpha})\), where \(\alpha> 0\) (the trace is nearly the exponential of half the length). Now this counting is with multiplicity. Also (same reference), the length spectrum determines the Riemann surface. Now in this paper Schmutz is counting merely the set of different traces. Each trace is counted once, no matter how often it occurs. The above theorem states that the growth of the counting function of this set is linear iff the group is arithmetic. In view of Huber's result then, arithmeticity is bound up with multiplicity in the length spectrum. As this paper carefully discusses, this Theorem was presaged by earlier work of \textit{K. Takeuchi} [J. Math. Soc. Japan 29, 91-106 (1977; Zbl 0344.20035) and ibid. 27, 600-612 (1975; Zbl 0311.20030)] as well as of \textit{W. Luo} and \textit{P. Sarnak} [Comm. Math. P…
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 922158
author: Paul Schmutz Schaller
reviewed by: Mark Sheingorn
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08407-0
OpenAlex ID: W1484848569
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description: scientific article; zbMATH DE number 922158