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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.

MSC:

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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