Related papers: Abelian tropical covers

We study whether a given tropical curve $\Gamma$ in $\mathbb{R}^n$ can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $\Gamma$. We give an affirmative answer to this…

We explore connections between the category of tropical abelian varieties (tav), $\mathbb{T}\mathcal{A}$, and the the category of tropical curves, $\mathbb{T}\mathcal{C}$, first in a broader context and then specifically by studying the…

We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in…

We define and investigate the tropical Prym varieties associated to unramified Galois cyclic covers of tropical curves (or equivalently metric graphs) $\tilde{\Gamma}\to \Gamma$. Our approach here is to study the tropical Prym varieties…

\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we…

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. It was shown earlier that the splice diagram determines the universal abelian cover of the manifold. We will in this article…

Given a tropical fan curve $\Sigma$ and a family of algebraic curves $X \rightarrow \mathbb{A}^k$ we define the realization locus $\mathop{Real}_\Sigma \subseteq \mathbb{A}^k$ as the set of fibers $X_a$ whose tropicalization is $\Sigma$. We…

Given a finite connected simple graph $\Gamma$, and a subgroup $G$ of its automorphism group, a general method for finding all finite abelian regular coverings of $\Gamma$ that admit a lift of each element of $G$ is developed. As an…

Abelian covers of hyperbolic $3$-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic $3$-manifolds. We obtain a classification theorem for measures invariant under the…

Let $\Gamma$ be the fundamental group of a finite connected graph $\mathcal G$. Let $\mathfrak M$ be an abelian group. A {\it distribution} on the boundary $\partial\Delta$ of the universal covering tree $\Delta$ is an $\mathfrak M$-valued…

Let $C$ be an affine curve over an algebraically closed field $k$ of characteristic $p>0$. Given an embedding problem $(\beta:\Gamma\longrightarrow G, \alpha: \pi^{et}_1(C)\longrightarrow G)$ for $\pi_1^{et}(C)$ where $\beta$ is a…

This paper gives a new explicit construction of the $\mathbb{Q}$-algebraic hull for virtually solvable groups $\Gamma$ of finite abelian ranks, taking into account the spectrum $S$ of the group $\Gamma$. As an application, we make a…

Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They…

A discrete subgroup $\Gamma$ of a locally compact group $H$ is called a uniform lattice if the quotient $H/\Gamma$ is compact. Such an $H$ is called an envelope of $\Gamma$. In this paper we study the problem of classifying envelopes of…

We show that, given a finitely generated group $G$ as the coordinate group of a finite system of equations over a torsion-free hyperbolic group $\Gamma$, there is an algorithm which constructs a cover of a canonical solution diagram. The…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…

Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2…

This paper is the third installment in a series of papers devoted to the computation of enumerative invariants of abelian surfaces through the tropical approach. We develop a pearl diagram algorithm similar to the floor diagram algorithm…

We prove that the tropical Abel--Prym map $\Psi\colon \widetilde\Gamma\to Prym(\widetilde\Gamma/\Gamma)$ associated with a free double cover $\pi\colon \widetilde\Gamma\to \Gamma$ of hyperelliptic metric graphs is harmonic of degree $2$ in…

This article applies the technical framework developed in previous work by the author to discrete admissible covers and their moduli spaces. More precisely, we construct a poic-space that parameterizes the discrete admissible covers after…