Hurwitz problem

Abstract: For a finite group G, the Hurwitz problem is to determine the genera of the surfaces which admit an effective G-action. (For a closed surface, genus means the number of handles, in the orientable case, or the number of crosscaps, in the nonorientable case. For a punctured or bounded surface, it means the algebraic genus, which is the rank of the fundamental group as a free group, or equivalently the first Betti number of the surface.) When the surface has negative Euler characteristic, various uniformization theorems allow one to assume that the action is conformal with respect to some Riemannian structure, dianalytic with respect to some dianalytic structure (i.e. Kleinian structure, see [A-G]), or isometric with respect to some complete hyperbolic structure with geodesic boundary (see Chap. 13 of [T1]). When the surfaces are Riemann surfaces, the problem is often considered for conformal (hence orientation-preserving) actions, but in the present work we make no restriction on orientability of the surface or the action. Although there are some results which concern the "s table" Hurwitz problem the determination up to finitely many cases of the genera admitting G-actions [K], [M-M] most attention has been focused on the determination of the minimal genus (g>2) admitting a G-action. Hurwitz [H2] found the general

Abstract: In 1923 there appeared a paper of A. Hurwitz, where he solved the problem of finding all the pairs of positive integers (n,p) and all systems of c jα k e ℝ, j, k=1, …,n; α=1,…,p, p≤n, such that the set of bilinear forms ηj=xα c jα k yk satisfied the condition Σj η j 2 = Σα x α 2 Σk y k 2 . The present authors reformulate the Hurwitz problem and its solution in the language of Clifford algebras, introducing the notion of the Hurwitz pair of two real unitary vector spaces V and S. Then they study the dependence of these pairs on induced symplectic decomposition and arrive in a natural way at a complex geometry induced by a given symplectic decomposition in connection with distinguishing a fixed direction in S, what introduces a kind of anisotropy.