Realization of finite Abelian groups by nets in P^2

On line arrangements with applications to 3-nets

We show a one-to-one correspondence between arrangements of d lines in the projective plane, and lines in P^{d-2}. We apply this correspondence to classify (3,q)-nets over the complex numbers for all q<=6. When q=6, we have twelve possible combinatorial cases, but we prove that only nine of them are realizable. This new case shows several new properties for 3-nets: different dimensions for moduli, strict realization over certain fields, etc. We also construct a three dimensional family of (3,8)-nets corresponding to the Quaternion group.

Realization of digraphs in Abelian groups and its consequences

Let $\overrightarrow{G}$ be a directed graph with no component of orderless than~$3$, and let $Γ$ be a finite Abelian group such that $|Γ|\geq 4|V(\overrightarrow{G})|$ or if $|V(\overrightarrow{G})|$ is large enough with respect to an arbitrarily fixed $\varepsilon>0$ then $|Γ|\geq (1+\varepsilon)|V(\overrightarrow{G})|$. We show that there exists an injective mapping $φ$ from $V(\overrightarrow{G})$ to the group $Γ$ such that $\sum_{x\in V(C)}φ(x) = 0$ for every connected component $C$ of $\overrightarrow{G}$, where $0$ is the identity element of $Γ$. Moreover we show some applications of th…

Realization of groups with pairing as Jacobians of finite graphs

We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur a…

Topological realizations and fundamental groups of higher-rank graphs

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor, and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely gen…

局部有限阿贝尔群中子集和密度的实现

Subset-Sum Density Realization in Locally Finite Abelian Groups

本文研究可数局部有限阿贝尔群中，通过有限子群滤过定义子集和集合的密度极限点，证明当 $|2G|=\infty$ 时任意区间可实现，并给出正密度下的加法闭包性质。

On minimal complements in groups

Let $W,W'\subseteq G$ be nonempty subsets in an arbitrary group $G$. The set $W'$ is said to be a complement to $W$ if $WW'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. We show that, if $W$ is finite then every complement of $W$ has a minimal complement, answering a problem of Nathanson. This also shows the existence of minimal $r$-nets for every $r\geqslant 0$ in finitely generated groups. Further, we give necessary and sufficient conditions for the existence of minimal complements of a certain class of infinite subsets in finitely generated abelian groups, partial…