Covering Transformation Group Research Articles

Enumeration of the BranchedmZp-Coverings of Closed Surfaces

In this paper, we enumerate the equivalence classes of regular branched coverings of surfaces whose covering transformation groups are the direct sum of m copies of Zp, p prime.

Zeta Functions for Images of Graph Coverings by Some Operations

We consider the zeta functions of the line graph and the middle graph of a regular covering of a graph G, and their related topics. Let M (G) be the middle graph of G and T (G) the total graph of G. We show that the middle graph and the total graph of a regular covering of a graph G with covering transformation group A is a regular covering of M (G) and T (G) with the same covering transformation group A, respectively. For a regular graph G, we express the zeta functions of the line graph and the middle graph of a regular covering of G by using the characteristic polynomial of that regular covering.

Enumeration of Regular Graph Coverings Having Finite Abelian Covering Transformation Groups

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors. An enumeration of the isomorphism classes of n-fold coverings of a graph G was done by Kwak and Lee [Canad. J. Math., XLII (1990), pp. 747--761] and independently by Hofmeister [Discrete Math., 98 (1991), pp. 437--444]. An enumeration of the isomorphism classes of connected n-fold coverings of a graph G was recently done by Kwak and Lee [J. Graph Theory, 23 (1996), pp. 105--109]. But the enumeration of the isomorphism classes of regular coverings of a graph G has been done for only a few cases. In fact, the isomorphism classes of ${\cal A}$-coverings of G were enumerated when ${\cal A}$ is the cyclic group $\BZ_n$, the dihedral group $\BD_n$ (n: odd), and the direct sum of m copies of $\BZ_p$. (See [Discrete Math., 143 (1995), pp. 87--97], [J. Graph Theory, 15 (1993), pp. 621--627], and [Discrete Math., 148 (1996), pp. 85--105]). In this paper, we discuss a method to enumerate the isomorphism classes of connected ${\cal A}$-coverings of a graph G for any finite group ${\cal A}$ and derive some formulas for enumerating the isomorphism classes of regular n-fold coverings for any natural number n. In particular, we calculate the number of the isomorphism classes of ${\cal A}$-coverings of G when ${\cal A}$ is a finite abelian group or the dihedral group $\BD_n$. Our method gives partial answers to the open problems 1 and 2 in [Discrete Math., 148 (1996), pp. 85--105] and also gives a formula to calculate the number of the subgroups of a given index of any finitely generated free abelian group.

Distribution of branched ¢D p-coverings of surfaces

A well-known theorem of Alexander [1] says that every orientable surface is a branched covering of the sphere S 2, and every nonorientable surface is a branched covering of the projective plane. In the study of surface branched coverings [2,3,14], we can ask naturally as a generalization of Alexander's theorem: In how many different ways can a given surface be a branched covering of another given surface? As a partial answer of this question, Kwak et al., [10] recently enumerated the equivalence classes of regular branched prime-fold coverings p : S → S for any given surfaces S and S . In this paper, we aim to enumerate the equivalence classes of the regular branched surface coverings p : S → S whose covering transformation group is the dihedral group D p of order 2 p, p prime. In some sense, this gives a classification of the pseudo-free D p-actions on a surface when a data for a quotient surface is given.

On the homotopy invariance of 𝐿² torsion for covering spaces

We prove the homotopy invariance of L 2 L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L 2 L^2 cohomology of the covering space vanishes, the homotopy invariance was established by Lück. We also give some applications of our results.

Regular graph coverings whose covering transformation groups have the isomorphism extension property

Enumerative results are presently a major center of interest in topological graph theory, as in the work of Gross and Furst [1], Hofmeister [5,6], Kwak and Lee [9–13] and Mull et al. [15], etc. Kwak and Lee [9] enumerated the isomorphism classes of graph bundles and those of n-fold graph coverings with respect to a group of automorphisms of the base graph which fix a spanning tree. Hofmeister [6] enumerated independently the isomorphism classes of n-fold graph coverings with respect to the trivial automorphism group of the base graph. But the enumeration of isomorphism classes of regular graph coverings has not been answered completely. As its partial answers, Hofmeister enumerated the isomorphism classes of Z 2-coverings (double coverings) with respect to any group of automorphisms of the base graph, and Sato [14] did the same work for Z p -coverings (regular prime-fold coverings). With respect to the trivial automorphism group of the base graph, Hong and Kwak [8] did the same work for Z 2 ⊕ Z 2 or Z 4-coverings, and Kwak and Lee [10] did it for Z p , Z p ⊕ Z 1 ( p ≠ q primes) or Z p 2 -coverings. As an expansion of this effort, we obtain in this paper several new algebraic characterizations for isomorphic regular coverings and derive an enumerating formula for the isomorphism classes of A-coverings of a graph G with respect to any group of automorphisms of G which fix a spanning tree, when the covering transformation group A has the isomorphism extension property. By definition, it means that every isomorphism between any two isomorphic subgroups B 1 and B 2 of A can be extended to an automorphism of A. Also, we obtain complete numerical enumeration of the isomorphism classes of Z n -coverings for all n, D n -coverings for odd n (D n is the dihedral group of order 2 n) or Z p ⊕ Z p -coverings of a graph G for prime p with respect to the trivial automorphism group of G. In addition, we applied our results to a bouquet of circles.

Equivariant minimax and minimal surfaces in geometric three-manifolds

Minimal surfaces in Riemannian three-dimensional manifolds have played a major role in recent studies of the geometry and topology of 3-manifolds. In particular, stable and least area minimal surfaces have been used extensively [SY, MSY, HS]. On the other hand, explicit examples of unstable minimal surfaces have rarely been given. For the 3-sphere S with the standard metric of constant sectional curvature, there is the classical paper of Lawson [LH], showing that closed orientable surfaces of every genus occur as (unstable) minimal surfaces embedded in 5 3 . More recently, Karcher, Pinkall, and Sterling [KPS] have constructed several new examples in S. We have discovered new infinite families of minimal surfaces in S. More generally, we announce a number of new finite and infinite families of embedded minimal surfaces in geometric 3-manifolds, using the minimax procedure described below. Geometric structures on 3-manifolds were introduced by Thurston [TW]. (See also the excellent survey by Scott [SP1].) There are eight geometries: R , S , S x R, Nil, H 2 x R, S L ^ R ) , Solv, and H . A geometric structure on a 3-manifold S is a representation of E as a quotient of one of the above eight spaces divided out by a covering transformation group acting isometrically. Equivalently, E is locally isometric to one of these spaces, with its natural homogeneous space structure. The first six of these geometries give Seifert fiber spaces and we are mainly interested in such examples. The SO(2)-isometry actions associated with most Seifert fiber spaces yield infinite classes of embedded minimal surfaces. Note that interesting examples can also be obtained in the other two geometries; e.g., in hyperbolic geometry H 3 [PR3, §2]. The minimax procedure has proved to be a versatile and powerful means of constructing unstable minimal surfaces in 3-manifolds. For basic details of this technique, see [PJ, SS, P R l , and PR2]. Suppose that G is a finite group of isometries acting on a closed oriented Riemannian 3-manifold E. Assume that A is a Heegaard surface in E; i.e., the closures of the components of E ~ A are handlebodies K and K'. Assume furthermore that A is G-equivariant; i.e., gA = A for all g E G. We consider one-parameter smooth families At, t € [0,1], sweeping out E and having the following properties: Ao and Ai are graphs; At is isotopic to A for all 0 < t < 1; At is G-equivariant for all t; the handlebody Kt is chosen so that the orientation on At is induced from that on

Projective structures on Riemann surfaces and Kleinian groups

Let S be a compact Riemann surface of genus 2, and let iv: U--->S be a holomorphic universal covering of S with the covering transformation group F , where U is the upper half plane {z e C : 1m z>0}. T h en , is a finitely generated Fuchsian group of the first kind on U and consists of hyperbolic Mobius transformations. We denote by B,(L, r) the Banach space of all holomorphic quadratic differentials for F defined on the lower half plane L . Namely, B A , F ) is the set of all holomorphic functions 0 on L satisfying

Three dualities on the integral homology of infinite cyclic coverings of manifolds

We consider a compact oriented topological n-manifold M. Let y be an element of the first integral cohomology H}(M) and M be the infinite cyclic covering space of M associated with y. The covering transformation group is infinite cyclic and denoted by with a generator t, specified by y. A subboundary, A, of M is 0 or a compact (n—l)-submanifold of the boundary dM such that A'=cldM(dM—A) is 0 or a compact (n—l)-submanifold of 3M. The pair (Af A') is called a splitting of dM. Let A be the lift of A, i.e., the preimage of A under the covering 1V[-+ M. Let Λ be the integral group ring of • The integral homology H*(M, A) forms a finitely generated Λ-module, because by [K/S] (M, A) is homotopy equivalent to a compact polyhedral pair and Λ is Noetherian. For an abelian group Hy let e H=Ext^(H; Z) (so that eH=Q for />2 and Homz(H,Z)= 3. The following Λ-submodule, DH} of H was introduced by Blanchfield [B]:

Notes on covering transformation groups

Notes on covering transformation groups