In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$ . We prove that the space of degree d real branched coverings having “many” real branched points (for example, more than $\sqrt {d}^{1+\alpha }$ , for any $\alpha>0$ ) has exponentially small measure. In particular, maximal real branched coverings – that is, real branched coverings such that all the branched points are real – are exponentially rare.

We prove that the determination of all $M^*$-groups is essentially equivalent to the determination of finite groups generated by an element of order 3 and an element of order 2 or 3 that admit a particular automorphism. We also show how the second commutator subgroup of an $M^*$-group $G$ can often be used to construct $M^*$-groups which are direct products with $G$ as one factor. Several applications of both methods are given.

AMS 2000 Mathematics subject classification: Primary 20D45; 20E36. Secondary 14H37; 30F50

The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M$_g/R$ of nonsingular real algebraic curves of genus g (g[les ]2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M$_g/R$. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M$_g/R$is real analytic. We conclude that M$_g/R$ is not a real analytic variety.