## 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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