Compact Riemann surfaces with many systoles
type: Item
instance of: scholarly article
title: Compact Riemann surfaces with many systoles
zbMATH Open document ID: 0867.30029
DOI: 10.1215/S0012-7094-96-08406-9
published in: Duke Mathematical Journal
publication date: 1997-08-04T00:00:00Z
review text: A systole is a minimal length simple closed geodesic on a surface. Here is an extract from the author's introduction: ``In [Math. Z. 223, No. 1, 13-25 (1996; reviewed below)]the author proved the following theorem: Let \(K\) be a positive integer. Then there exists a Riemann surface \(M_K\) corresponding to a principal congruence subgroup of \(\text{PSL} (2,\mathbb{Z})\) so that \(M_K\) has more systoles than \(K\cdot \dim (T(M_K))\) where \(T(M_K)\) is the Teichmüller space of \(M_K\). \(\dots\) [of course,] these surfaces all have cusps. The case of compact surfaces is treated in this paper. I shall prove the following theorem. Theorem. Let \(K\) be a positive integer. Let \(p\equiv 3\pmod 4\) be a prime. Then there exists a closed Riemann surface \(M_K\) of genus \(g\) corresponding to a congruence subgroup of the Fuchsian arithmetic group derived from the quaternion algebra \({(p,-1) \over\mathbb{Q}}\) so that \(M_K\) has more systoles than \(K(6g-6)\). Notice that the size of the automorphism group of these surfaces is not responsible for the big number of systoles since this size is bounded by \(14(6g-6)\) (this is Hurwitz's theorem). Therefore, this result is rather surprising. For genus 2, 3, 4, and 5, for example, there are no closed surfaces known with more than \(12g-12\) systoles, and they probably do not exist; compare the author [Geom. Funct. Anal. 3, 564-631 (1993; Zbl 0810.53034)] and [Manuscripta Math. 85, 429-447 (1994; Zbl 0819.30027)]. The proof of the theorem needs four steps. First, a way to classify the different systoles has to be found. It will be sufficient to consider only a part of the systoles, namely, those which contain a fixed point of an involution of the surface. I shall call them canonical systoles. The second step is the construction of as many different canonical systoles as one needs. The automorphism group of the surface acts on the systoles and induces a partition of the systoles in isometry classes. The third step consists in…
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 922157
zbMATH Keywords: Fuchsian arithmetic group
zbMATH Keywords: systoles
author: Paul Schmutz Schaller
reviewed by: Mark Sheingorn
MaRDI profile type: Publication
cites work: Products of conjugacy classes in groups
cites work: Q4039784
cites work: Q3975268
cites work: Q4004260
cites work: Arithmetic Fuchsian groups and the number of systoles
cites work: Congruence subgroups and maximal Riemann surfaces
cites work: Riemann surfaces with shortest geodesic of maximal length
cites work: Systoles on Riemann surfaces
cites work: Q5625265
cites work: Arithmétique des algèbres de quaternions
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description: scientific article; zbMATH DE number 922157