Semi-arithmetic Fuchsian groups and modular embeddings. (English) Zbl 0968.11022

Schmutz Schaller, Paul; Wolfart, Jürgen

J. Lond. Math. Soc., II. Ser. 61, No. 1, 13-24 (2000).

A modular embedding of a Fuchsian group \(\Gamma\) operating on the Poincaré upper half-plane \({\mathcal H}\) is a group inclusion \(f:\Gamma \to \Delta\) and a holomorphic embedding \(F: {\mathcal H} \to {\mathcal H}^r\), where \(r\) is a suitable integer, and \(\Delta\) is an arithmetic group, such that \(F(\gamma z) = f(\gamma) F(z)\) for all \(\gamma \in \Gamma\). J. Wolfart and P. B. Cohen [Acta Arith. 56, 93-110 (1990; Zbl 0717.14014)] showed that every Fuchsian triangle group admits a modular embedding. Few of the triangle groups are arithmetic; however, they are what the authors here call semi-arithmetic.
A Fuchsian group \(\Gamma\) is semi-arithmetic if the traces of the elements of the subgroup generated by the squares in \(\Gamma\) are algebraic integers, all lying in some finite degree extension of the rationals. The authors show that any Fuchsian group which admits a modular embedding is semi-arithmetic. They exhibit infinitely many examples of semi-arithmetic groups of a particular signature; show that there are many semi-arithmetic groups which do not admit modular embeddings; refine the types of semi-arithemtic groups.
The paper concludes with the remark that since Fuchsian groups which admit modular embeddings can be shown to uniformize Riemann surfaces which are arithmetic, a conjecture of D. Chudnovsky and G. Chudnovsky [Computers and mathematics, Proc. Int. Conf., Stanford/CA (USA) 1986, Lect. Notes Pure Appl. Math. 125, 109-232 (1990; Zbl 0712.11078)] predicts that all such groups are either arithmetic or finite index subgroups of triangle groups.

Reviewer: Thomas Schmidt (Corvallis)

MSC:

11F06	Structure of modular groups and generalizations; arithmetic groups
20H10	Fuchsian groups and their generalizations (group-theoretic aspects)

Keywords:

arithmetic groups; triangle groups; uniformization; modular embedding; Fuchsian group