On Abelian Branched Coverings of the Sphere

 We obtain an enumeration formula for the number of weak equivalence classes of the branched (?×ℬ)-covering of the sphere with m-branch points, when ? and ℬ are finite abelian groups with (|?|,|ℬ|)=1. From this, we can deduce an explicit formula for enumerating the weak equivalence classes of pseudofree spherical (ℤ _p ×ℤ _q )-actions on a given surface, when p and q are distinct primes. Received: August 10, 1999 Final version received: June 19, 2000     

Branched Coverings and Embedded Surfaces in Four-manifolds

In this paper, we get a lower bound on the genera of surfaces re, presenting certain divisible classes in some 4-manifold X with Hi（X; Z） finite.     

Representing Dehn Twists with Branched Coverings

We show that any homologically nontrivial Dehn twist of a compactsurface F with boundary is the lifting of a half-twist in thebraid group , with respectto a suitable branched covering p : F B². In particular, weallow the surface to have disconnected boundary. As a consequence,any allowable Lefschetz fibration on B² is a branched coveringof B² x B².     

The Complexity of 2-Fold Branched Coverings of a 3-Sphere

Complexity is a function which maps a set of 3-manifolds to a set of nonnegative integers. This function has some natural properties (finiteness, additivity, etc.) and it shows, in some sense how complex the manifold is. This function seems important for the classification of 3-manifolds. We evaluate of the complexity of 2-fold branched coverings of a 3-sphere. We present a theoretical estimate and compare it with experimental data.     

Determining Knotsw and Links by Cyclic Branched Coverings

It is well known that different knots or links in the 3-sphere can have homeomorphic n-fold cyclic branched coverings. We consider the following problem: for which values of nis a knot of link determined by itsn-fold cyclic branched covering? We consider the class of hyperbolic resp.2π/n-hyperbolic links. The isometry or symmetry groups of such links are finite, and their n-fold branched coverings are hyperbolic 3-manifolds. Our main result states that if ndoes not divide the order of the finite symmetry group of such a link, then the link is determined by its n-fold branched covering. In a sense, the result is best possible; the key argument of its proof is algebraic using some basic result about finite p-groups. The main result applies, for example, to the cyclic branched coverings of the 2-bridge links; in particular, it gives a classification of the maximally symmetricD₆-manifolds which are exactly the 3-fold branched coverings of the 2-bridge links.     

On Surface Attractors and Repellers in 3-Manifolds

We show that if f: M ³ → M ³ is an A diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ with support $M_\mathcal{B}^2$ , then $\mathcal{B} = M_\mathcal{B}^2$ and there exists a k ≥ 1 such that (1) $M_\mathcal{B}^2$ is the disjoint union M ₁ ² ? ? ? M _k ² of tame surfaces such that each surface M _i ² is homeomorphic to the 2-torus T ²; (2) the restriction of f ^k to M _i ² , i ∈ {1,..., k}, is conjugate to an Anosov diffeomorphism of the torus T ².     

Coverings of Convex Sets and Volume Inequalities

We give diverse coverings of convex sets and sections of convex sets in R ⁿ by balls and cylinder sets, and establish volume inequalities which generalize the famous Santalo and inverse Santalo inequalities. The tools which we use are developed from the viewpoint of geometric and functional analysis. Received May 27, 1998, and in revised form January 25, 1999.     

On v-Distal Flows on 3-Manifolds

In [2], H. Furstenberg studied a distal action of a locallycompact group G on a compact metric space X, and establisheda structure theorem. As a consequence, he showed that if G isabelian, then a simply connected space X does not admit a minimaldistal G-action. In this paper we concern ourselves with a nonsingular flow = {^t} on a closed 3-manifold M. Recall that is called distalif for any distinct two points x, y M, the distance d(^tx, ^ty)is bounded away from 0. The distality depends strongly uponthe time parametrization. For example, there exists a time parametrizationof a linear irrational flow on T² which yields a nondistal flow[4, 6]. 1991 Mathematics Subject Classification 58F25, 57R30.     

On the Hausdorff Dimension of Some Random Cantor Sets

We consider Cantor sets C _X,p constructed by means of a sequence {X _k ^–p } _k=1 of random variables. We find sufficient conditions for the inequality to hold.     

Branched Coverings of Cubical Complexes and Subgroups of Hyperbolic Groups

By considering branched coverings of piecewise Euclidean cubicalcomplexes, the paper provides an example of a torsion free hyperbolicgroup containing a finitely presented subgroup which is nothyperbolic.     

On 2-Systoles of Hyperbolic 3-Manifolds

We investigate the geometry of π ₁-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any ${\epsilon > 0}$ , if the manifold M has sufficiently large systole sys₁(M), the genus of any such surface in M is bounded below by ${{\rm exp}((\frac{1}{2} - \epsilon){\rm sys}_1(M))}$ . Using this result we show, in particular, that for congruence covers M _i → M of a compact arithmetic hyperbolic 3-manifold we have: (a) the minimal genus of π ₁-injective surfaces satisfies ${{\rm log} \, {\rm sysg}(M_i) \gtrsim \frac{1}{3} {\rm log} \, {\rm vol}(M_i) ; (b)}$ there exist such sequences with the ratio Heegard ${{\rm genus}(M_i)/{\rm sysg}(M_i) \gtrsim {\rm vol}(M_i)^{1/2}}$ ; and (c) under some additional assumptions π ₁(M _i ) is k-free with ${{\rm log} \, k \gtrsim \frac{1}{3}{\rm sys}_1(M_i)}$ . The latter resolves a special case of a conjecture of Gromov.     

Isometry Groups of Hyperbolic Three-Manifolds which are Cyclic Branched Coverings

Let M be a hyperbolic three-manifold which is an n-fold cyclic branched covering of a hyperbolic link L in the three-sphere, or more precisely, of the hyperbolic three-orbifold whose underlying topological space is the three-sphere and whose singular set, of branching index n, the link L. We say that M has no hidden symmetries (with respect to the given branched covering) if the isometry group of M is the lift of (a subgroup of) the isometry group of the hyperbolic orbifold (which is isomorphic to the symmetry group of the link L). It follows from Thurston's hyperbolic surgery theorem that M has no hidden symmetries if n is sufficiently large. Our main result is an explicit numerical version of this fact: we give a constant, in terms of the volume of the complement of L, such that M has no hidden symmetries for all n larger than this constant; we show by examples that a universal constant working for all hyperbolic knots or links does not exist. We give also some results on the possible orders and the structure of the isometry group of M. Finally, we construct sets of four different -hyperbolic knots which have the same two-fold branched covering (a hyperbolic three-manifold); it is an interesting question for how many different -hyperbolic knots (or links) this may happen (in the case of hyperbolic knots, for arbitrarily many).     

On Central Cantor Sets with Self-Arithmetic Difference of Positive Lebesgue Measure

We prove the existence of C^r - (but not C^r+1-) regular centralCantor sets with zero Lebesgue measure such that their selfarithmetic difference is a Cantor set with positive Lebesguemeasure. This is motivated by a conjecture in the field of bifurcationsof dynamical systems posed by Jacob Palis.     

On a Class of Contact Riemannian 3-Manifolds

In this paper we classify 3-dimensional complete contact Riemannian manifolds satisfying the following condition “the characteristic vector field is an eigenvector of the Ricci operator with constant eigenvalue”. Moreover we prove that this condition is equivalent with some other characteristic ones.     

Algebraic Topology for Minimal Cantor Sets

It will be shown that every minimal Cantor set can be obtained as a projective limit of directed graphs. This allows to study minimal Cantor sets by algebraic topological means. In particular, homology, homotopy and cohomology are related to the dynamics of minimal Cantor sets. These techniques allow to explicitly illustrate the variety of dynamical behavior possible in minimal Cantor sets. submitted 20/07/05, accepted 18/10/05