This master’s thesis studies the systole and orthosystole of Riemann surfaces. Given a notion of lengths of curves in a surface, the systole (resp. orthosystole) corresponds to the infimum of lengths of essential closed (resp. proper) curves. We will mostly consider the hyperbolic and extremal systoles. In particular, we will compute their value for a highly symmetric family of surfaces, the regular ideal polygons. We will then show that these surfaces maximize the orthosystole among ideal polygons with the same number of sides and that the same result generalizes to systoles of double polygons.

This master’s thesis studies the Laplace operator on Riemann surfaces. We are especially interested in its eigenvalues, which correspond to the notes that the surface would play if it were a drum. In particular, the first non-zero eigenvalue λ1 and its multiplicity m1 (the dimension of the corresponding eigenspace) have been well studied. For instance, Colin de Verdière conjectured that m1 is bounded above by the chromatic number minus 1 based on a few examples. Later work by Fortier Bourque and Petri showed that among hyperbolic surfaces of genus 3, the Klein quartic maximizes the multiplicity, and attains the upper bound conjectured by Colin de Verdière. This surface is the first of a sequence of highly symmetrical surfaces named Hurwitz surfaces. We will show using the Selberg trace formula that for the next surface in the sequence, the Fricke–Macbeath surface F, we have m1(F) = 7. This result was also obtained independently by Chul-hee Lee. Chapter 1 introduces some geometric notions including hyperbolic geometry, hyperbolic surfaces, and triangular surfaces, followed by Hurwitz’s automorphism theorem. Chapter 2 covers some basic concepts in spectral theory as well as some useful tools like the Selberg trace formula and a bit of representation theory. Chapter 3 focuses on the study of the Fricke–Macbeath surface and the proof of our main result using the techniques introduced in previous chapters. Finally, Chapter 4 discusses new methods for computing m1 which were used to show the existence of counterexamples to Colin de Verdière’s conjecture in joint work with Fortier Bourque, Gruda-Mediavilla, and Petri.