On a Riemann surface S of finite type containing a family of N disjoint disks Di (“islands”), we consider several natural conformal invariants measuring the distance from the islands to ∂S and separation between different islands. In a near degenerate situation we establish a relation between them called the QuasiAdditivity Law. We then generalize it to a Quasi-Invariance Law providing us with ...

Regular map is an algebraic concept to describe most symmetric tilings of closed surfaces of arbitrary genus. All regular maps resp. symmetric tilings of surfaces up to genus 302 are algebraically known in the form of symmetry groups acting on their universal covering spaces. But still little is known about geometric realizations, i.e. finding most symmetric embeddings of closed surfaces and a ...

A (generalized) weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps. If the covering maps are regular, then we call the inverse limit space a strong solenoid. By a theorem of M.C. McCord, strong solenoids are homogeneous. We show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids. We also give an example of a w...

The stabilizer of a fixed point class of a map is the fixed subgroup of the induced fundamental group homomorphism based at a point in the class. A theorem of Jiang, Wang and Zhang is used to prove that if a map of a graph satisfies a strong remnant condition, then the stabilizers of all its fixed point classes are trivial. Consequently, if φp,f is the nvalued lift to a covering space p of a ma...

A continuous map between topological f : X → Y is said to satisfy the path-lifting property if for any path p : [0, 1] → Y and any x ∈ f(p(0)) there exists a lifting p̃ of the path p with the intitial value x, i.e. there exists a path p̃ such that f ◦ p̃ = p and p̃(0) = x. Similarly, a smooth map between Riemannian manifolds f : X → Y is said to satisfy the rectifiable path-lifting property if the ...

The canonical covering maps from Hurwitz varieties to configuration varieties are important in algebraic geometry. The schemetheoretic fiber above a rational point is commonly connected, in which case it is the spectrum of a Hurwitz number field. We study many examples of such maps and their fibers, finding number fields whose existence contradicts standard mass heuristics.

We define two classes of topological infinite degree covering maps modeled on two families of transcendental holomorphic maps. The first, which we call exponential maps of type (p, q), are branched covers and is modeled on transcendental entire maps of the form Pe, where P and Q are polynomials of degrees p and q. The second is the class of universal covering maps from the plane to the sphere w...

In this paper we study the topological class of universal covering maps from the plane to the sphere with two removed points; we call the elements topological transcendental maps with two asymptotic values and we denote the space byAV2. We prove that an element f ∈ AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian deri...

In this paper we continue to study (‘strong’) Nielsen coincidence numbers (which were introduced recently for pairs of maps between manifolds of arbitrary dimensions) and the corresponding minimum numbers of coincidence points and pathcomponents. We explore compatibilities with fibrations and, more specifically, with covering maps, paying special attention to selfcoincidence questions. As a sam...

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C∗-algebras C(X) ⋊α,L N introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated b...