# zbMATH — the first resource for mathematics

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

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