A structured description of the genus spectrum of abelian $p$-groups

On the genus of the commuting graphs of finite non-abelian groups

The commuting graph of a non-abelian group is a simple graph in which the vertices are the non-central elements of the group, and two distinct vertices are adjacent if and only if they commute. In this paper, we classify (up to isomorphism) all finite non-abelian groups whose commuting graphs are acyclic, planar or toroidal. We also derive explicit formulas for the genus of the commuting graphs of some well-known class of finite non-abelian groups, and show that, every collection of finite non-abelian groups whose commuting graphs have the same genus is finite.

阿贝尔$p$-群的亏格谱的新方法

A new approach to the genus spectra of abelian $p$-groups

针对$p>2$的阿贝尔$p$-群，提出一种新方法，通过可有限步计算的函数给出亏格谱的结构描述。

On Kahler extensions of abelian groups

We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. The main tools come from surface topology and known restrictions on Kahler groups.

Genus of commuting conjugacy class graph of groups

For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute the genus of $\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, tor…

几乎局部自由群与亏格问题

Almost locally free groups and the genus question

Sacerdote [Sa]证明了非阿贝尔自由群在一阶语言L $_o$中精确地满足适用于群论的相同的普遍存在句Th（F $_2$）$\cap \forall \exists $。证明了在Th（$_2$）$\cap \forall \exists $的每个模型中，极大Abel子群初等等价于局部循环群（必然非平凡且无挠）.在非Abel局部自由群和Remeslennikov的$\exists $ -自由群之间插入两类群.这些类是\textbf{almost locally free groups}和\textbf{quasi-locally free groups}。特别地，几乎局部自由% \textbf{\ }群是Th（F $_2$）$\cap \forall \exists $的模型，而准局部自由群是具有极大阿贝尔子群的$\exists $ -自由群，它们初等等价于局部循环群（必然非平凡且无挠）.两个主要的悬而未决的问题在光谱的两端是：（1）。是不是每一个几乎本地生成的自由群都是自由的？（2.）是不是每个拟局部自由群都几乎局部自由？在Abelianizations中包含非平凡挠的拟局部自由群的例子比比皆是。几乎局部自由群是否具有无挠阿贝尔化的问题与自由群中将导出群的某些元素表示为矩阵的乘积所需的因子数的界有关。

具有环面子群格的有限阿贝尔群

Finite Abelian Groups with Toroidal Subgroup Lattices

研究确定了多类阿贝尔群的子群格亏格，完成了子群格可嵌入环面的全部有限阿贝尔群的分类工作。