Abelian Cover Research Articles

Elementary Abelian Covers of Graphs

Let \cal CG(X) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G ≤ Aut X of automorphisms lifts. There is a natural lattice structure on \cal CG(X), where ℘1 ≤ ℘2 whenever ℘2 factors through ℘1. The sublattice \cal CG(℘) of coverings which are below a given covering ℘ : X˜ → X naturally corresponds to a lattice \cal NG(℘) of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications. For a given prime p, let \cal CGp(X) ≤ \cal CG(X) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs \cal CGp(X) in the sense that, for each member of \cal CGp(X), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.

The rank of elliptic surfaces in unramified abelian towers

Let ℰ → C be an elliptic surface defined over a number field K. For a finite covering C′ → C defined over K, let ℰ′ = ℰ ×CC′ be the corresponding elliptic surface over C′. In this paper we give a strong upper bound for the rank of ℰ′ (C′/K) in the case of geometrically abelian unramified coverings C′ → C and under the assumption that the Tate conjecture is true for ℰ′/K. In the case that C is an elliptic curve and the map C′ = C → C is the multiplication-by-n map, the bound for rank(ℰ′(C′/K)) takes the form O(nk/log logn), which may be compared with the elementary bound of O(n2).

Universal abelian covers of quotient-cusps

The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a ℚ-Gorenstein singularity.

Linear criteria for lifting automorphisms of elementary abelian regular coverings

For a given finite connected graph Γ, a group H of automorphisms of Γ and a finite group A, a natural question can be raised as follows: Find all the connected regular coverings of Γ having A as its covering transformation group, on which each automorphism in H can be lifted. In this paper, we investigate the regular coverings with A= Z p n , an elementary abelian group and get some new matrix-theoretical characterizations for an automorphism of the base graph to be lifted. As one of its applications, we classify all the connected regular covering graphs of the Petersen graph satisfying the following two properties: (1) the covering transformation group is isomorphic to the elementary abelian p-group Z p n , and (2) the group of fibre-preserving automorphisms of a covering graph acts arc-transitively. As a byproduct, some new 2- and 3-arc-transitive graphs are constructed.

Mahler measure, links and homology growth

Let l be an oriented link of d components with nonzero Alexander polynomial Δ( u 1,…, u d ). Let Λ be a finite-index subgroup of H 1(S 3−l)≅ Z d , and let M Λ be the corresponding abelian cover of S 3 branched along l. The growth rate of the order of the torsion subgroup of H 1( M Λ ), as a suitable measure of Λ approaches infinity, is equal to the Mahler measure of Δ.

Bounds in groups with finite abelian coverings or with finite derived groups

Bounds in groups with finite abelian coverings or with finite derived groups

Abelian coverings of regular hypermaps of types {4,4,4} and {3,3,4} of genus 2

In this paper. we obtain elementary abelian coverings of the regular hypermaps of genus 2 corresponding to the orientation-preserving automorphisın groups .$L2(3 ), the speciai linear group of order 24. and OK . the quaternion group of order 8. We also determine the reflexibility of these coverings.

Cocycles and stable foliations of Axiom A flows

We show that the strong stable foliation of the canonical extension of an Axiom A flow to an Abelian cover of a compact manifold is ergodic with respect to the product of a Gibbs measure on the manifold times the Haar measure on the fiber, as soon as it is transitive on the support of the measure. This generalizes previous results of M. Babillot, F. Ledrappier, M. Pollicott and V. Kaimanovich.

Holomorphic Functions of Exponential Growth on Abelian Coverings of a Projective Manifold

Let M be a projective manifold, p: MG→M a regular covering over M with a free Abelian transformation group G. We describe the holomorphic functions on MG of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary, we obtain Cartwright and Liouville-type theorems for such functions. Our approach brings together the L2 cohomology technique for holomorphic vector bundles on complete Kähler manifolds and the geometric properties of projective manifolds.

FIRST BETTI NUMBERS OF ABELIAN COVERINGS OF THE COMPLEX PROJECTIVE PLANE BRANCHED OVER LINE CONFIGURATIONS

We prove the following results concerning the first Betti numbers of abelian coverings of CP2 branched over line configurations: (1) An estimate of the first Betti numbers. (2) A characterization of central and general position line configurations in terms of the first Betti numbers of abelian coverings. (3) The first Betti numbers of the abelian coverings of the real line configurations up to 7 components.

Von Neumann Betti numbers and Novikov type inequalities

In this paper we show that Novikov type inequalities for closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a consequence we obtain a vanishing theorem for L2 cohomology. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings.

Jacobian Tori Associated with a Finite Graph and Its Abelian Covering Graphs

We develop a graph-theoretic analogue of the Jacobian torus, which is defined, for a finite graph X, as the torus H1(X,R)/H1(X,Z) with a natural flat metric. It is observed that this notion, especially the volume of Jacobian tori, appears in many contexts of graph theory.

Fundamental Group for Some Cuspidal Curves

We construct a family of plane curves as pull-backs of a conic for abelian coverings of P^2. If the conic is tangent to the ramification lines one obtains a family of curves of degree 2n with 3n singularities of type A_{n-1}. We calculate the fundamental group and Alexander polynomial for any member of this family and for some deformations of it.

ON THE COMPUTATION OF THE TURAEV-VIRO MODULE OF A KNOT

Let M be the manifold obtained by 0-framed surgery along a knot K in the 3-sphere. A Topological Quantum Field Theory assigns to a fundamental domain of the universal abelian cover of M an operator, whose non-nilpotent part is the Turaev-Viro module of K. In this paper, using surgery formulas, we give a matrix presentation for the Turaev-Viro module of any knot K, in the case of the (Vp, Zp) TQFT of Blanchet, Habegger, Masbaum and Vogel. We do the computation for a family of knots in the special case p = 8, and note the relation with the fibering question.

Commutator coverings of Siegel threefolds

We investigate the existence and non-existence of modular forms of low weight with a character with respect to the paramodular group $\Gamma_t$ and discuss the resulting geometric consequences. Using an advanced version of Maa\ss lifting one can construct many examples of such modular forms and in particular examples of weight 3 cusp forms. Consequently we find many abelian coverings of low degree of the moduli space ${\Cal A}_t$ of (1,t)-polarized abelian surfaces which are not unirational. We also determine the commutator subgroups of the paramodular group $\Gamma_t$ and its degree 2 extension $\Gamma^+_t$. This has applications for the Picard group of the moduli stack ${\underline{\Cal A}}_t$. Finally we prove non-existence theorems for low weight modular forms. As one of our main results we obtain the theorem that the maximal abelian cover ${\Cal A}_t^{com}$ of ${\Cal A}_t$ has geometric genus 0 if and only if t=1, 2, 4 or 5. We also prove that ${\Cal A}_t^{com}$ has geometric genus 1 for t=3 and 7.

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.

Horospheres on abelian covers

We consider the strong stable foliation of the geodesic flow for a noncompact, connected abelian cover of a closed negatively curved manifold. We show that there exists proper leaves, and that non-proper leaves are dense.

Holomorphic functions of polynomial growth on abelian coverings of a compact complex manifold

Holomorphic functions of polynomial growth on abelian coverings of a compact complex manifold

On the Homology of Finite Abelian Coverings of Links

AbstractLet A be a finite abelian group and M be a branched cover of an homology 3-sphere, branched over a link L, with covering group A. We show that H1(M; Z[1/|A|]) is determined as a Z[1/|A|][A]-module by the Alexander ideals of L and certain ideal class invariants.

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group GMis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ⋛ 2. A Galois cover f :X ↪ Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.