Compact Riemann surfaces with many systoles (Q1922423)

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 showing that an isometry class of systoles contains only a bounded number of canonical systoles so that an unbounded number of different canonical systoles induces an unbounded number of different isometry classes of systoles. Since these isometry classes are big enough (this is the fourth step), the theorem follows.'' The surfaces constructed are quotients of \(\Gamma_{(p,-1)} (N)\), Fuchsian groups defined as follows. The surfaces constructed are quotients of \(\Gamma_{(p,-1}(N)\), Fuchsian groups defined as follows: For every even integer \(N\geq 2\) and every prime \(p\equiv 3 \pmod 4\), define \[ \Gamma_{(p,-1)} (N)= \left\{ \left[ \begin{matrix} 1+N(a+b \sqrt p) & N(-c+d \sqrt p) \\ N(c+d \sqrt p) & 1+N(a-b \sqrt p) \end{matrix} \right];\;a,b,c,d\in \mathbb{Z} \right\}, \] where the determinant of the matrices has to be 1. The construction is explicit, a nice combination of number theory and algebra. This fine paper is very clearly written.