A geometric characterization of arithmetic Fuchsian groups

Abstract

The trace set of a Fuchsian group $\Gamma$ encodes the set of lengths of closed geodesics in the surface $\Gamma \backslash \mathbb{H}$. Luo and Sarnak [3] showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering (BC) property. Sarnak [5] then conjectured that the BC property actually characterizes arithmetic Fuchsian groups. Schmutz [6] stated the even stronger conjecture that a cofinite Fuchsian group is arithmetic if its trace set has linear growth. He proposed a proof of this conjecture in the case when the group $\Gamma$ contains at least one parabolic element, but unfortunately, this proof contains a gap. In this article, we point out this gap, and we prove Sarnak's conjecture under the assumption that the Fuchsian group $\Gamma$ contains parabolic elements.