Abelian Cover Research Articles

The special value 𝑢=1 of Artin-Ihara 𝐿-functions

We study the special value u = 1 u=1 of Artin-Ihara L L -functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest.

The taut polynomial and the Alexander polynomial

AbstractLandry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmüller polynomial of a fibred face of the Thurston norm ball. We prove that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. Thus, the Teichmüller polynomials are just specialisations of twisted Alexander polynomials. We also give formulae relating the taut polynomial and the untwisted Alexander polynomial. There are two formulae, depending on whether the maximal free abelian cover of a veering triangulation is edge‐orientable or not. Furthermore, we consider 3‐manifolds obtained by Dehn filling a veering triangulation. In this case, we give formulae that relate the specialisation of the taut polynomial under a Dehn filling and the Alexander polynomial of the Dehn‐filled manifold. This extends a theorem of McMullen connecting the Teichmüller polynomial and the Alexander polynomial to the non‐fibred setting, and improves it in the fibred case. We also prove a sufficient and necessary condition for the existence of an orientable fibred class in the cone over a fibred face of the Thurston norm ball.

Holomorphic polylogarithms and Bloch complexes

Abstract For an integer n ≥ 2 {n\geq 2} we define a polylogarithm ℒ ^ n {\widehat{\mathcal{L}}_{n}} , which is a holomorphic function on the universal abelian cover of ℂ ∖ { 0 , 1 } {\mathbb{C}\setminus\{0,1\}} defined modulo ( 2 ⁢ π ⁢ i ) n / ( n - 1 ) ! {(2\pi i)^{n}/(n-1)!} . We use the formal properties of its functional relations to define groups ℬ ^ k ⁢ ( F ^ ) {\widehat{\mathcal{B}}_{k}(\widehat{F})} lifting Goncharov’s Bloch groups ℬ k ⁢ ( F ) {\mathcal{B}_{k}(F)} of a field F, and show that they fit into a complex Γ ^ ⁢ ( F , n ) {\widehat{\Gamma}(F,n)} lifting Goncharov’s Bloch complex Γ ⁢ ( F , n ) {\Gamma(F,n)} . When F = ℂ {F=\mathbb{C}} , we show that the imaginary part (when n is even) or real part (when n is odd) of ℒ ^ n {\widehat{\mathcal{L}}_{n}} agrees with a real single valued polylogarithm ℒ n {\mathcal{L}_{n}} on the group H 1 ⁢ ( Γ ^ ⁢ ( ℂ , n ) ) {H^{1}(\widehat{\Gamma}(\mathbb{C},n))} . When n = 2 {n=2} , this group is Neumann’s extended Bloch group. Goncharov’s complex conjecturally computes the rational motivic cohomology of F, and one may speculate whether the extended complex computes the integral motivic cohomology. Finally, we use ℒ ^ 3 {\widehat{\mathcal{L}}_{3}} to construct a lift of Goncharov’s map H 5 ⁢ ( SL ⁡ ( 3 , ℂ ) ) → ℝ {H_{5}(\operatorname{SL}(3,\mathbb{C}))\to\mathbb{R}} to a complex valued map whose real part agrees with that of Goncharov. The lift makes use of the cluster ensemble structure on the Grassmannian Gr ⁡ ( 3 , 6 ) {\operatorname{Gr}(3,6)} .

Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor. Furthermore, he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface. Subsequently, he studies the same theme for Enriques surfaces.

Free representations of outer automorphism groups of free products via characteristic abelian coverings

Abstract Given a free product 𝐺, we investigate the existence of faithful free representations of the outer automorphism group Out ⁡ ( G ) \operatorname{Out}(G) , or in other words of embeddings of Out ⁡ ( G ) \operatorname{Out}(G) into Out ⁡ ( F m ) \operatorname{Out}(F_{m}) for some 𝑚. This is based on work of Bridson and Vogtmann in which they construct embeddings of Out ⁡ ( F n ) \operatorname{Out}(F_{n}) into Out ⁡ ( F m ) \operatorname{Out}(F_{m}) for some values of 𝑛 and 𝑚 by interpreting Out ⁡ ( F n ) \operatorname{Out}(F_{n}) as the group of homotopy equivalences of a graph 𝑋 of genus 𝑛, and by lifting homotopy equivalences of 𝑋 to a characteristic abelian cover of genus 𝑚. Our construction for a free product 𝐺, using a presentation of Out ⁡ ( G ) \operatorname{Out}(G) due to Fuchs-Rabinovich, is written as an algebraic proof, but it is directly inspired by Bridson and Vogtmann’s topological method and can be interpreted as lifting homotopy equivalences of a graph of groups. For instance, we obtain a faithful free representation of Out ⁡ ( G ) \operatorname{Out}(G) when G = F d ∗ G d + 1 ∗ ⋯ ∗ G n G=F_{d}\ast G_{d+1}\ast\cdots\ast G_{n} , with F d F_{d} free of rank 𝑑 and G i G_{i} finite abelian of order coprime to n - 1 n-1 .

Cyclic and Abelian coverings of real varieties

We describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties.

Classification of Arc-Transitive Elementary Abelian Covers of the C13 Graph

Let Γ be a graph and G⩽Aut(Γ). A graph Γ can be called G-arc-transitive (GAT) if G acts transitively on its arc set. A regular covering projection p:Γ¯→Γ is arc-transitive (AT) if an AT subgroup of Aut(Γ) lifts under p. In this study, by applying a number of concepts in linear algebra such as invariant subspaces (IVs) of matrix groups (MGs), we discuss regular AT elementary abelian covers (R-AT-EA-covers) of the C13 graph.

On the Prym map of Galois coverings

We consider the Prym variety P(C˜∕C) associated to a Galois coverings of curves f:C˜→C branched at r points. We discuss some properties and equivalent definitions and then consider the Prym map 𝒫=𝒫(G,g,r):R(G,g,r)→Ap,δ with δ the type of the polarization. For Galois coverings whose Galois group is abelian and metabelian (nonabelian) we show that the differential of this map at certain points is injective. We also consider the Abel-Prym map u:C˜→P(C˜∕C) and prove some results for its injectivity. In particular, we show that in contrast to the classical and cyclic case, the behavior of this map here is more complicated. The theories of abelian and metabelian Galois coverings play a substantial role in our analysis and have been used extensively throughout the paper.

Orbifold splice quotients and log covers of surface pairs

A three-dimensional orbifold $(\Sigma, \gamma_i, n_i)$, where $\Sigma$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $\mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with the case where $C$ is empty and one considers the universal abelian cover, under certain conditions on a resolution graph one can construct pairs and their universal abelian log covers. Such pairs are called orbifold splice quotients.

Moving Seshadri constants, and coverings of varieties of maximal Albanese dimension

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian etale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian etale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.

Long Time Asymptotics of Heat Kernels and Brownian Winding Numbers on Manifolds with Boundary

Let M be a compact Riemannian manifold with smooth boundary. We obtain the exact long time asymptotic behaviour of the heat kernel on abelian coverings of M with mixed Dirichlet and Neumann boundary conditions. As an application, we study the long time behaviour of the abelianized winding of reflected Brownian motions in M. In particular, we prove a Gaussian type central limit theorem showing that when rescaled appropriately, the fluctuations of the abelianized winding are normally distributed with an explicit covariance matrix.

Graph covers with two new eigenvalues

A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the Sensitivity Conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this connection is a natural first example of the relationship between group valued adjacency matrices with few eigenvalues, and combinatorially interesting covering graphs. In particular, we define a two-eigenvalue cover, to be a covering graph whose adjacency spectra differs (as a multiset) from that of the graph it covers by exactly two eigenvalues. We show that walk regularity of a graph implies walk regularity of any abelian two-eigenvalue cover. We also give a spectral characterization for when a cyclic two-eigenvalue cover of a strongly-regular graph is distance-regular.

Optimal results in Lorentzian Aubry\u2013Mather theory

This article complements the Lorentzian Aubry–Mather Theory in Suhr (Geom Dedicata 160:91–117, 2012; J Fixed Point Theory Appl 21:71, 2019) by giving optimal multiplicity results for the number of maximal invariant measures. As an application the optimal Lipschitz continuity of the time separation on the Abelian cover is established.

Abelian Covers and Non-Commuting Sets in a Non-Abelian p-Group Which its Central Quotient is Metacyclic

Let G be a group. A set S in G is said to be non-commuting if $$xy \ne yx$$ for any two distinct elements $$x,y \in S$$ . We define w(G) to be the maximum possible cardinality of a non-commuting set in G. In this paper, we determine w(G) for a finite non-abelian p-group G such that G/Z(G) is metacyclic by obtaining an abelian centralizers cover of this group. As a consequence, we show that the set of all commuting automorphisms of a finite non-abelian p-group G such that G/Z(G) is metacyclic, forms a subgroup of $$\mathrm {Aut}(G)$$ .

Enumerating Abelian Typical Cube-Free Fold Coverings of a Circulant Graph

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. A covering projection p from a Cayley graph Cay(Γ, X) onto another Cayley graph Cay(Q, Y) is called typical if the function p : Γ → Q on the vertex sets is a group epimorphism. A typical covering is called abelian (or circulant, respectively) if its covering graph is a Cayley graph on an abelian (or a cyclic, respectively) group. Recently, the equivalence classes of connected abelian typical prime-fold coverings of a circulant graph are enumerated. As a continuation of this work, we enumerate the equivalence classes of connected abelian typical cube-free fold coverings of a circulant graph.

Minimal Boundaries in Tonelli Lagrangian Systems

Abstract We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, that is, a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $\textrm{T} M$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Mañé critical value of the universal abelian cover. This provides in particular a class of nonreversible Finsler metrics on the two-sphere possessing infinitely many closed geodesics.

Taut foliations in branched cyclic covers and left-orderable groups

We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology 3 3 -spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible 3 3 -manifolds containing a genus one fibred knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibred knot in an integer homology 3 3 -sphere admits a co-oriented taut foliation and has left-orderable fundamental group, even if the surgered manifold does not, and that the same holds for many branched covers of satellite knots with braided patterns. A key fact used in our proofs is that the Euler class of a universal circle representation associated to a co-oriented taut foliation coincides with the Euler class of the foliation’s tangent bundle. Though known to experts, no proof of this important result has appeared in the literature. We provide such a proof in the paper.

A new example of an algebraic surface with canonical map of degree 16

In this note, we construct a minimal surface of general type with geometric genus $$ p_g =4 $$ , self-intersection of the canonical divisor $$ K^2 = 32$$ , and irregularity $$ q = 1 $$ such that its canonical map is an Abelian cover of degree 16 of $$\mathbb P^1\times \mathbb P^1$$ .

On the averages of generalized Hasse–Witt invariants of pointed stable curves in positive characteristic

In the present paper, we study fundamental groups of curves in positive characteristic. Let $$X^{\bullet }$$ be a pointed stable curve of type $$(g_{X}, n_{X})$$ over an algebraically closed field of characteristic $$p>0$$, $$\Gamma _{X^{\bullet }}$$ the dual semi-graph of $$X^{\bullet }$$, and $$\Pi _{X^{\bullet }}$$ the admissible fundamental group of $$X^{\bullet }$$. In the present paper, we study a kind of group-theoretical invariant $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ associated to the isomorphism class of $$\Pi _{X^{\bullet }}$$ called the limit of p-averages of $$\Pi _{X^{\bullet }}$$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, we give a lower bound and a upper bound of $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$. In particular, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ under a certain assumption concerning $$\Gamma _{X^{\bullet }}$$ which generalizes a formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ obtained by Tamagawa. Moreover, if $$X^{\bullet }$$ is a component-generic pointed stable curve, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian etale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.

On class A Lorentzian 2-tori with poles II: Foliations by timelike lines

In this paper, we show that if (T2,g) is a class A Lorentzian 2-torus with timelike poles, then there exists a Lipschitz foliation by complete future-directed timelike geodesics with any pre-assigned asymptotic direction in the interior of the stable time cone. This is done by constructing certain C1,1 solutions to the equation g(∇u,∇u)=−1 on the Abelian cover (R2,g).