A geometric characterization of arithmetic Fuchsian groups. (English) Zbl 1141.20029

Let \(\Gamma\) be a Fuchsian group, that is, a discrete subgroup of \(\text{PSL}(2,\mathbb{R})\). The trace set of \(\Gamma\) is defined as \(\text{Tr}(\Gamma)=\{\text{tr}(\gamma)\mid\gamma\in\Gamma\}\). We say that \(\text{Tr}(\Gamma)\) satisfies the bounded clustering (BC) property if and only if there exists a constant \(B(\Gamma)\) such that \(\text{Tr}(\Gamma)\cap[n,n+1]\) has less than \(B(\Gamma)\) elements for all integers \(n\).
If \(\Gamma\) is a cofinite Fuchsian group with parabolic elements then \(\text{Tr}(\Gamma)\) satisfies the BC property if and only if \(\Gamma\) is arithmetic. This beautiful result proves Sarnak’s conjecture for the case that \(\Gamma\) contains parabolic elements.


20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
11F06 Structure of modular groups and generalizations; arithmetic groups
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
22E40 Discrete subgroups of Lie groups
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