Let Γ be a non-Euclidean crystallographic group. Γ is said to be non-maximal if there exists a non-Euclidean crystallographic group Γ such that Γ Γ and the dimension of the Teichmüller space of Γ equals the dimension of the Teichmüller space of Γ . The full list of such pairs of groups is computed in the case when Γ is non-normal in Γ . The corresponding problem for Fuchsian groups was solved by Singerman.

Abstract. The moduli space M g of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of the orbifold is the branch locus B g . In this article we show that B g is connected for genera three, four, thirteen, seventeen, nineteen and fiftynine, and disconnected for any other genus. In order to prove this we use Fuchsian groups, automorphisms of order 5 and 7 of Riemann surfaces, and calculations with GAP for some small genera.

Abstract. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces. Harvey [9] alluded to the existence of the equisymmetric stratification of the moduli space M g of Riemann surfaces of genus g, each strata consists in the points of the moduli space corresponding to equisymmetric surfaces. The branch locus B g of M g is formed by the strata corresponding to surfaces of genus g admitting nontrivial automorphisms (or admitting other automorphisms that are the identity and the hyperelliptic involution if g = 2). Broughton [5] showed that the equisymmetric stratification is indeed a stratification of M g by irreducible algebraic subvarieties whose interior, if it is non-empty, is a smooth, connected, locally closed algebraic subvariety of M g , Zariski dense in the stratum. In this way we can equip the moduli space with a structure of complex of groups.It is well known that B 1 consists of two points and B 2 is not connected, since R. Kulkarni (see [11]) showed that the curve w 2 = z 5 − 1 is isolated in B 2 , i.e. this single surface is a connected component of B 2 . More precisely B 2 has exactly two connected components (see [1]). On the contrary the branch locus B 3 is connected (see also [1]).

We determine, for all genus g ≥ 2 the Riemann surfaces of genus g with 4g automorphisms. For g = 3, 6, 12, 15 or 30, this surfaces form a real Riemann surface F g in the moduli space M g : the Riemann sphere with three punctures. The set of real Riemann surfaces in F g consists of three intervals its closure in the Deligne-Mumford compactification of M g is a closed Jordan curve.

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