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Existence and non-existence of torsion in maximal arithmetic Fuchsian groups. (English) Zbl 1195.20053

The author continues his study of torsion in maximal arithmetic Fuchsian groups. The results in this paper are of two kinds. On the one hand, they substantiate the prediction that \(2\)-torsion is prevalent. On the other hand, the author constructs specific examples of torsion-free maximal arithmetic Fuchsian groups. In particular, it is shown that in every commensurability class of arithmetic Fuchsian groups, there are infinitely many conjugacy classes of maximal groups which contain elements of order \(2\). Every maximal arithmetic Fuchsian group defined over \(\mathbb{Q}\) has \(2\)-torsion.
Let \(k\) be a totally real number field and let \(A\) be a quaternion division algebra over \(k\) such that it is ramified at all but one real place of \(k\). If \(\mathcal O\) is an order in \(A\), and \(N(\mathcal O^+)\), the elements of the normalizer which have totally positive norm, then the projection of \(N(\mathcal O^+)\) at the unramified real place is an arithmetic Fuchsian group.
An example of a torsion-free maximal arithmetic Fuchsian group is the following one. Let \(k=\mathbb{Q}(\alpha)\) where \(\alpha^3-4\alpha-1=0\); this has class number \(1\) and has narrow class number \(2\). In this case a maximal order \(\mathcal O\) can be so chosen that the corresponding maximal arithmetic Fuchsian group is torsion-free.

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)
11R52 Quaternion and other division algebras: arithmetic, zeta functions
16H10 Orders in separable algebras
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