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

TL;DR

This paper proves Sarnak and Schmutz's conjecture that the trace set of a non-uniform Fuchsian lattice grows linearly if and only if it is arithmetic, and analyzes the volume growth of cocompact lattice embeddings.

Contribution

It confirms the conjecture for non-uniform lattices and analyzes the volume of lattice embeddings with super-polynomial trace set growth.

Findings

Trace set of non-uniform lattice exhibits linear growth iff lattice is arithmetic

Cocompact lattice embeddings with trace set growth exceeding n^{2-ε} have positive Weil-Petersson volume

Asymptotic volume analysis of lattice embedding sets

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 non-uniform lattice, we prove Scmutz'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 of genus bigger than 2 and any positive number $\epsilon$, te set of cocompact lattice embedding such that their growth rate of trace set exceeds $n^{2-\epsilon}$ has positive Weil-Petersson volume. We also provide an asymptotic analysis of the volume of this set.