We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold b...

We prove that the Eynard-Orantin symplectic invariants of the curve xy − y2 = 1 are the orbifold Euler characteristics of the moduli spaces of genus g curves. We do this by associating to the Eynard-Orantin invariants of xy − y2 = 1 a problem of enumerating covers of the two-sphere branched over three points. This viewpoint produces new recursion relations—string and dilaton equations—between t...

Consider the Hurwitz space parameterizing covers of P branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmüller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmüller curves, the sum of Lyapunov exponents and the Siegel-Veech const...

of the branched covers of an elliptic curve. Here, N (m) g,d is the (weighted) number of isomorphism classes of branched covers, with genus g(> 1), degree d, and ramification index (m,m, . . . ,m), of an elliptic curve. Such a cover is called an m-simple cover. Our aim is to prove that the formal power series F (m) g converges to a function belonging to the graded ring of quasi-modular forms wi...

Motivated by the local lifting problem for Galois covers of curves, this paper investigates Galois branched covers of the open p-adic disc. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive...

Motivated by the local lifting problem for Galois covers of curves, this paper investigates Galois branched covers of the open p-adic disc. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is completely determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications...

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of the branched covers counted by the Hurwitz numbers, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzyks...

Let k be an algebraically closed field of characteristic p > 0. In this note we consider Galois covers g : Y → P1k which are only branched at t = ∞. We call such covers unramified covers of the affine line. It follows from Abhyankar’s conjecture for the affine line proved by Raynaud ([7]) that such a cover exists for a given group G if and only if G is a quasi-p group, i.e. can be generated by ...