Hurwitz components of groups with socle PSL(3, q)

For a finite group G, the Hurwitz space Hinr,g(G) is the space of genus g covers of the Riemann sphere P1 with r branch points and the monodromy group G. In this paper, we give a complete list of some almost simple groups of Lie rank two. That is, we assume that G is a primitive almost simple groups of Lie rank two. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in Hinr,g(G).

Double Hurwitz Numbers and Multisingularity Loci in Genus 0

In the Hurwitz space of rational functions on ${{\mathbb{C}}}\textrm{P}^1$ with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincaré dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.

The uniformization of the moduli space of principally polarized abelian 6-folds

Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_{6}-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.

Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7

The Hurwitz space is the space of genus covers of the Riemann sphere with branch points and the monodromy group . Let be the symmetric group . In this paper, we enumerate the connected components of . Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of . This work gives us the complete classification of primitive genus zero symmetric group of degree seven.

Connected Components of H_(r,g)^A (G)

The Hurwitz space is the space of genus g covers of the Riemann sphere with branch points and the monodromy group . In this paper, we enumerate the connected components of the Hurwitz spaces for a finite primitive group of degree 7 and genus zero except . We achieve this with the aid of the computer algebra system GAP and the MAPCLASS package.

VEECH GROUPS FOR HOLONOMY-FREE TORUS COVERS

For fixed coprime k, l ∈ ℕ and each pair (w, z) ∈ ℂ2we define an infinite cyclic cover Σk,l(w, z) → 𝕋, called a k-l-surface or k-l-cover. We show that [Formula: see text] classifies k-l-covers up to isomorphism away from a rather small set. The diagonal action of SL2(ℤ) on ℂ2descends to [Formula: see text], reflecting the SL2(ℤ)-action on the family of k-l-surfaces equipped with a translation structure. The moduli space of holonomy free k-l-surfaces is a compact SL2(ℤ) invariant subspace [Formula: see text] containing all k-l-surfaces with a lattice stabilizer with respect to the SL2(ℤ) action. We calculate the stabilizer, the Veech group, explicitly and represent k-l-covers branched over two points by a generalized class of staircase surfaces. Finally we study SL2(ℤ)-equivariant translation maps from the Hurwitz space of k-(d - k)-covers to Hurwitz spaces of ℤ/d-covers branched over two points.