Arithmetic groups and the length spectrum of Riemann surfaces
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. Phys. 161, 419-432 (1994; Zbl 0797.58069)]. (However, and as expected, proofs of previous results are generally not given.) The proof of the theorem uses a sophisticated explicit geometric treatment of Thurston pairs-of-pants, together with interesting symmetry techniques -- the author is probably without peer in these matters. The reviewer offers the following comment: This theorem opens the possibility that rather than viewing arithmetic and nonarithmetic as a dichotomy, perhaps it is proper to use the order of Schmutz's growth function as an index of how ``arithmetic'' a group is. This order is presumably between \(T\) and \(T^2\). Perhaps the nonarithmetic Hecke triangle groups have growth nearer to \(T\), for example. It seems to the reviewer that to obtain such results requires an original way of holding down the number different possible traces -- something other than demanding integer matrices, say. (The reviewer discussed this recently with the author, who was not optimistic.) But if it should turn out that there are groups with intermediate growth rates, this paper will have been a real turning point in the theory of arithmetic groups!
scientific article; zbMATH DE number 922158
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