On trace sets of hyperbolic surfaces and a conjecture of Sarnak and Schmutz

Abstract

In this paper, we investigate the trace set of a Fuchsian lattice. There are two results of this paper. The first is that for a nonuniform lattice, we prove Schmutz’s conjecture: the trace set of a Fuchsian lattice exhibits linear growth if and only if the lattice is arithmetic. Additionally, we show that for a fixed surface group ${\Gamma }_g$ of genus $g\ge 3$ and any $\epsilon >0$, the set of cocompact lattice embeddings such that their growth rate of the trace set exceeds $n^{2-\epsilon }$ has positive Weil–Petersson volume. We also provide an asymptotic analysis of the volume of this set as $g\to \infty$.