Covering Transformation Group Research Articles

Enumerating regular graph coverings whose covering transformation groups are $$\mathbb {Z}_p$$-extensions of a cyclic group

Enumerating regular graph coverings whose covering transformation groups are $$\mathbb {Z}_p$$-extensions of a cyclic group

Preimage homomorphism indices of preimage classes

Under finite regular covering spaces, each preimage class of the base spaces is covered by some preimage classes of the covering spaces. We find an explicit relationship between those preimage classes in terms of covering transformation groups. Thereby we obtain a relationship of the homomorphism indices between them in terms of covering transformation groups. This is a local averaging formula and is the main result of our paper. The local formula gives rise to a global formula for the preimage homomorphism index and a formula for the preimage Reidemeister trace.

On cubic symmetric non-Cayley graphs with solvable automorphism groups

It was proved in Feng et al. (2015) that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a 2-regular graph of type 22, that is, a graph with no automorphism of order 2 interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic 2-regular graphs of type 22 with a solvable automorphism group is constructed, and the smallest graph has order 6174. This answers a question posed by Estélyi and Pisanski in 2016. Moreover, it includes a subfamily of graphs which are connected 2-regular covers of the Pappus graph with covering transformation group Zp3, and these graphs were missed in Oh (2009).

Enumerating regular graph coverings whose covering transformation groups are ℤ_2-extensions of a cyclic group

Several types of the isomorphism classes of graph coverings have been enumerated by many authors. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n -fold coverings of a graph by the second and third authors. For regular coverings of a graph, their isomorphism classes were enumerated when the covering transformation group is a finite abelian or dihedral group. In this paper, we enumerate the isomorphism classes of graph coverings when the covering transformation group is a ℤ 2 -extension of a cyclic group, including generalized quaternion and semi-dihedral groups.

2-Arc-Transitive Regular Covers of Kn,n Having the Covering Transformation Group ℤp2

This paper contributes to the classification of finite 2-arc-transitive graphs. In [12], all the regular covers of complete bipartite graphs Kn,n were classified, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of Kn,n, whose covering transformation group is elementary abelian group of order p2 and whose fibre-preserving automorphism group acts 2-arc-transitively. As a result, two new infinite families of 2-arc-transitive graphs are found. Moveover, it will be explained that it seems to be infeasible to classify all such covers when the covering transformation group is an elementary abelian group of order pk for an arbitrary integer k.

Locally 3-Arc-Transitive Regular Covers of Complete Bipartite Graphs

In this paper, locally $3$-arc-transitive regular covers of complete bipartite graphs are studied, and results are obtained that apply to arbitrary covering transformation groups. In particular, methods are obtained for classifying the locally $3$-arc-transitive graphs with a prescribed covering transformation group, and these results are applied to classify the locally $3$-arc-transitive regular covers of complete bipartite graphs with covering transformation group isomorphic to a cyclic group or an elementary abelian group of order $p^2$.

2-ARC-TRANSITIVE REGULAR COVERS OF HAVING THE COVERING TRANSFORMATION GROUP

This paper contributes to the regular covers of a complete bipartite graph minus a matching, denoted $K_{n,n}-nK_{2}$, whose fiber-preserving automorphism group acts 2-arc-transitively. All such covers, when the covering transformation group $K$ is either cyclic or $\mathbb{Z}_{p}^{2}$ with $p$ a prime, have been determined in Xu and Du [‘2-arc-transitive cyclic covers of $K_{n,n}-nK_{2}$’, J. Algebraic Combin.39 (2014), 883–902] and Xu et al. [‘2-arc-transitive regular covers of $K_{n,n}-nK_{2}$ with the covering transformation group $\mathbb{Z}_{p}^{2}$’, Ars. Math. Contemp.10 (2016), 269–280]. Finally, this paper gives a classification of all such covers for $K\cong \mathbb{Z}_{p}^{3}$ with $p$ a prime.

2-Arc-Transitive regular covers of K_{n,n} - nK_2 with the covering transformation group Z_p^2

In 2014, Xu and Du classified all regular covers of a complete bipartite graph K n , n minus a matching, denoted by K n , n − n K 2 , whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of K n , n − n K 2 , whose covering transformation group is isomorphic to Z p 2 with p a prime and whose fibre-preserving automorphism group acts 2 -arc-transitively. Actually, there are only few covers with these properties and it is shown that all of them are covers of K 4, 4 − 4 K 2 .

Lifting graph automorphisms along solvable regular covers

A solvable cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of lifts of the automorphism in the layers of the abelian covers. This procedure is applied to classify metacyclic covers of the tetrahedron branched at face-centers.

Non-abelian almost totally branched coverings over the platonic maps

A map is a 2-cell embedding of a connected graph into a closed surface. A map is orientable if the supporting surface is orientable. An orientable map is regular if its group of orientation-preserving automorphisms acts transitively on the darts. Using an equivalent algebraic description of regular maps and their coverings, we employ the theory of group extensions to classify the almost totally branched coverings of the platonic maps with non-abelian covering transformation groups, generalising the results of Hu, Nedela and Wang.

Stability of pseudoconvexity of disc bundles over compact Riemann surfaces and application to a family of Galois coverings

It is proved that Galois coverings of smooth families of compact Riemann surfaces over Stein manifolds are holomorphically convex if the covering transformation groups are isomorphic to discrete subgroups of the automorphism group of the unit disc. The proof is based on an extension of the fact that disc bundles over compact Kähler manifolds are weakly 1-complete.

2-Arc-transitive metacyclic covers of complete graphs

Regular covers of complete graphs whose fibre-preserving automorphism groups act 2-arc-transitively are investigated. Such covers have been classified when the covering transformation groups K are cyclic groups Zd for an integer d≥2, metacyclic abelian groups Zp2, or nonmetacyclic abelian groups Zp3 for a prime p (see S.F. Du et al. (1998) [5] for the first two metacyclic group cases and see S.F. Du et al. (2005) [3] for the third nonmetacyclic group case). In this paper, a complete classification is achieved of all such covers when K is any metacyclic group.

Arc-Transitive Dihedral Regular Covers of Cubic Graphs

A regular covering projection is called dihedral or abelian if the covering transformation group is dihedral or abelian. A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs. In particular, we classify all arc-transitive dihedral regular covers of $K_4$, $K_{3,3}$, the 3-cube $Q_3$ and the Petersen graph.

2-Arc-transitive cyclic covers of K n,n −nK 2

Du et al. (in J. Comb. Theory B 74:276---290, 1998 and J. Comb. Theory B 93:73---93, 2005), classified regular covers of complete graph whose fiber-preserving automorphism group acts 2-arc-transitively, and whose covering transformation group is either cyclic or isomorphic to $\mathbb{Z}_{p}^{2}$ or $\mathbb{Z}_{p}^{3}$ with p a prime. In this paper, a complete classification is achieved of all the regular covers of bipartite complete graphs minus a matching K n,n ?nK 2 with cyclic covering transformation groups, whose fiber-preserving automorphism groups act 2-arc-transitively.

Lifting automorphisms along abelian regular coverings of graphs

This article proposes an effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group.

Non-ultra regular digital covering spaces with nontrivial automorphism groups

The study of digital covering transformation groups (or automorphism groups, discrete deck transformation groups) plays an important role in the classification of digital spaces (or digital images). In particular, the research into transitive or nontransitive actions of automorphism groups of digital covering spaces is one of the most important issues in digital covering and digital homotopy theory. The paper deals with the problem: Is there a digital covering space which is not ultra regular and has an automorphism group which is not trivial? To solve the problem, let us consider a digital wedge of two simple closed ki-curves with a compatible adjacency, i ?{1,2}, denoted by (X, k). Since the digital wedge (X, k) has both infinite or finite fold digital covering spaces, in the present paper some of these infinite fold digital covering spaces were found not to be ultra regular and further, their automorphism groups are not trivial, which answers the problem posed above. These findings can be substantially used in classifying digital covering spaces and digital images so that the paper improves on the research in Section 4 of [3] (compare Figure 2 of the present paper with Figure 2 of [3]), which corrects an error that appears in the Boxer and Karaca's paper [3] (see the points (0,0), (0,8), (6, -1) and (6,7) in Figure 2 of [3]).

Complement to explicit description of Hopf surfaces and their automorphism groups

In the previous paper we determined Aut(X) of each Hopf surface X W G with W C 2 (0, 0) so that its holomorphic automorphism group is given by Aut(X) Aut(X) G. We calculate the group of connected components 0(Aut(X)) by reviewing the classification. A Hopf surface X is a compact complex surface whose universal covering space is W C 2 (0, 0). So, X W G by denoting its covering transformation group by G. A Hopf surface is called primary if its fundamental group G is isomorphic to the group Z of integers, and secondary otherwise. In Theorem 1 of [3] we determined Aut(X) of each secondary Hopf surface X W G so that its holomorphic automorphism group is given by Aut(X) Aut(X) G, where Aut(X) is the normalizer of G in the holomorphic automorphism group of C 2 fixing (0, 0). Moreover, in the following cases (2) and (3) it co incides with the normalizer of G in GL(2, C). We calculate 0(Aut(X)) including primary Hopf surfaces as a continuation by correcting some parts of [3]. Before stating Theorem 2 we review the classification of Hopf surfaces. Except the special case (0) that G is not given by any subgroup of GL(2, C), we may assume that G GL(2, C) and G is an extension of H g G det g 1 by Z. Let a be

Weighted zeta functions for quotients of regular coverings of graphs

Let G be a connected graph. We reformulate Stark and Terras' Galois Theory for a quotient H of a regular covering K of a graph G by using voltage assignments. As applications, we show that the weighted Bartholdi L-function of H associated to the representation of the covering transformation group of H is equal to that of G associated to its induced representation in the covering transformation group of K. Furthermore, we express the weighted Bartholdi zeta function of H as a product of weighted Bartholdi L-functions of G associated to irreducible representations of the covering transformation group of K. We generalize Stark and Terras' Galois Theory to digraphs, and apply to weighted Bartholdi L-functions of digraphs.

The end curve theorem for normal complex surface singularities

We prove the “End Curve Theorem,” which states that a normal surface singularity (X, o) with rational homology sphere link Σ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function (X, o) → (ℂ, 0) whose zero set intersects Σ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” (X, o) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in ℂ\_t\_ , where t is the number of leaves in the resolution graph for (X, o), together with an explicit description of the covering transformation group. Among the immediate consequences of the End Curve Theorem are the previously known results: (X, o) is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005).

On fixed points of automorphisms of non-orientable unbordered Klein surfaces

In 1973, Macbeath found a general formula for the number of points fixed by an arbitrary orientation preserving automorphism of a Riemann surface X. It was given in terms of a group G of conformal automorphisms of X and the ramification data of the covering X --> X/G, which corresponds to the so called universal covering transformation group. In these terms, for the case of a cyclic group of automorphisms of an unbordered non-orientable Klein surface, the formula was given later by Izquierdo and Singerman and here we find formulas valid for an arbitrary (finite) group G of automorphisms.