Length coordinates for moduli spaces and the distribution of semi-arithmetic hyperbolic surfaces

Abstract

This thesis investigates the interplay between hyperbolic geometry, moduli space, and arithmetic structures on surfaces, with a focus on semi-arithmetic and subarithmetic hyperbolic surfaces. On the geometric side, we show that any closed hyperbolic surface of genus g ≥ 2 in the thick part of moduli space is determined by the lengths of at most 12g − 12 closed geodesics, with explicit logarithmic bounds in g. This provides a quantitative form of length-based determination, optimal up to multiplicative constants. We apply this framework to arithmetic problems. Within the class of arithmetic surfaces, we obtain polynomial lower bounds for the Teichmüller distance between distinct finite coverings of a fixed surface in this class. In the semi-arithmetic setting, we prove counting results for surfaces with bounded invariants, showing that their number grows at most superexponentially with the genus. Finally, we establish a denseness result for subarithmetic surfaces of signature (1, 1), proving that subarithmetic Q–pieces form a dense subset of the corresponding moduli space.