The trace set of a Fuchsian group encodes the set of lengths of closed geodesics in the surface . Luo and Sarnak  showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering (BC) property. Sarnak  then conjectured that the BC property actually characterizes arithmetic Fuchsian groups. Schmutz  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 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 contains parabolic elements.
"A geometric characterization of arithmetic Fuchsian groups." Duke Math. J. 142 (1) 111 - 125, 15 March 2008. https://doi.org/10.1215/00127094-2008-002