We consider holomorphic maps f : U → U for a hyperbolic domain U in the complex plane, such that the iterates of f converge to a boundary point ζ of U . By a previous result of the authors, for such maps there exist nice absorbing domains W ⊂ U . In this paper we show that W can be chosen to be simply connected, if f has parabolic I type in the sense of the Baker–Pommerenke–Cowen classification...

Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2,C) Isom(H3). The volume of ρ is defined by taking any ρ-equivariant map from the universal cover W̃ to H and then by integrating the pullback of the hyperbolic volume form on a fundamental domain. This volume does not depend on the choice of the equivariant map because two equivariant maps are always eq...

We give a form of the Van Kampen Theorem involving covering morphisms in a lextensive category. This includes the usual results for covering maps of locally connected spaces, for light maps of compact Hausdorff spaces, and for locally strong separable algebras. @ 1997 Elsevier Science B.V. 1991 Muth. Sut~j. Cluss.: lSB99,57MlO

We use an algebraic description of surfaces with boundary to study covering maps. The focus is on the relationship between automor-phisms in the base and lifted maps. We show how to introduce and/or prohibit additional automorphisms in the lift. Using this control we give proofs of MacBeath's Theorem, Gr unbaum's Conjecture, and settle a conjecture of Nedela and Skoviera.

In this paper we define the concept of a ramified covering map in the category of simplicial sets and we show that it has properties analogous to those of the topological ramified covering maps. We show that the geometric realization of a simplicial ramified covering map is a topological ramified covering map, and we also consider the relation with ramified covering maps in the category of simp...

We study the existence of geometrically controlled branched covering maps from R to open 3-manifolds or to decomposition spaces S/G, and from S/G to S. Dedicated to the memory of Fred Gehring, whose work has been our inspiration

In this paper we define, for each aspherical orientable 3-manifold M endowed with a torus splitting T , a 2-dimensional fundamental l1-class [M ] whose l1norm has similar properties as the Gromov simplicial volume of M (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of M and the l1-norm of [M ] to give a complete characteriz...

The primary intent of the present paper is to adapt familiar notions of algebraic topology, such as the fundamental group, homology, and cohomology to the context of algebraic maps and their (ramified) covering projections. The notion of a principal derived map is already fairly well understood in terms of the defining voltages; however, once it is recognized that voltages are essentially cohom...

Let M̃ → M be a holomorphic (unbranched) covering map between two compact complex manifolds, with b2(M̃) = 1. We prove that if M̃ and M both admit regular Kähler forms ω̃ and ω respectively then, up to homotheties, (M̃, ω̃) and (M, ω) are biholomorphically isometric.

We show the space of expanding Blaschke products on S is compactified by a sphere of invariant measures, reminiscent of the sphere of geodesic currents for a hyperbolic surface. More generally, we develop a dynamical compactification for the Teichmüller space of all measure-preserving topological covering maps of S.