A geometric characterization of arithmetic Fuchsian groups

Details A geometric characterization of arithmetic Fuchsian groups A geometric characterization of arithmetic Fuchsian groups

2006-09-17

arxiv.org/abs/math/0609477v1

Duke Math. J. 142 (2008), 111-125

Mathematics

Differential Geometry

23 pages, 4 figures

Scientific paper

The trace set of a Fuchsian group $\Gamma$ ist the set of length of closed geodesics in the surface $\Gamma \backslash \mathbb{H}$. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C 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 $\Gamma$ contains at least one parabolic element, but unfortunately this proof contains a gap. In the present paper we point out this gap and we prove Sarnak's conjecture under the assumption that the Fuchsian group $\Gamma$ contains parabolic elements.

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