By a construction of Berstein and Edmonds every proper branched cover f between manifolds is factor covering orbit map from locally connected compact Hausdorff space called the monodromy to target manifold. For covers 2-manifolds known be We show that this does not generalize dimension 3 by constructing self-map 3-sphere for which contractible space.

Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones...

This paper develops a harmonic Galois theory for finite graphs, thereby classifying harmonic branched G-covers of a fixed base X in terms of homomorphisms from a suitable fundamental group of X together with G-inertia structures on X. As applications, we show that finite embedding problems for graphs have proper solutions and prove a Grunwald-Wang type result stating that an arbitrary collectio...

Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones...

We study birational geometry of Fano varieties, realized as double covers σ: V → P M , M ≥ 5, branched over generic hypersurfaces W = W 2(M −1) of degree 2(M − 1). We prove that the only structures of a rationally connected fiber space on V are the pencils-subsystems of the free linear system | − 1 2 K V |. The groups of birational and biregular self-maps of the variety V coincide: Bir V = Aut V .

Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones...

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is uniqu...

Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones...

Examples suggest that there is a correspondence between L-spaces and 3manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-manifold that is non-hyperbolic, a family which includes all...

Hurwitz spaces are spaces of pairs S, f where S is a Riemann surface and f : S → ̂ C a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces Hp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of p − 1 /2 transpositions and the monodromy group is the dihedral group ...