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

Geninska, Slavyana; Leuzinger, Enrico

Duke Math. J. 142, No. 1, 111-125 (2008).

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.

Reviewer: Gerhard Rosenberger (Dortmund)

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

Keywords:

arithmetic Fuchsian groups; quaternion algebras; parabolic elements; discrete subgroups; bounded clustering property; cofinite Fuchsian groups