Metabelian圈积为LERF

Subgroup Distortion and the Relative Dehn Functions of Metabelian Groups

We show the connection between the relative Dehn function of a finitely generated metabelian group and the distortion function of a corresponding subgroup in the wreath product of two free abelian groups of finite rank. Further, we show that if a finitely generated metabelian group $G$ is an extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn function is bounded above by the exponential function up to equivalence.

某些亚阿贝尔群中的丢番图问题

Diophantine Problem in Some Metabelian Groups

该研究证明了Baumslag-Solitar群$BS(1,k)$与有限生成阿贝尔群和无限循环群的圈积中，二次方程的丢番图问题具有可判定性，拓展了群论中丢番图问题的可判定范围。

几类群的圈积的畸变

Distortion of wreath products in some finitely presented groups

圈积如Z wr Z不是有限可表示的，但可以作为有限可表示群的子群出现。在这里，我们计算Z wr Z作为Thompson群F的子群和Baumslag亚交换群G的子群的畸变。我们发现Z wr Z在F中是不失真的，但在G中至少是指数失真的。

亚阿贝尔群中测地线的逼近

Approximation of Geodesics in Metabelian Groups

已知自由亚阿贝尔群中的有界测地线长度问题是NP-完全的（特别地，测地线问题是NP-难的）。本文构造了一个求解测地线问题的2-近似多项式时间确定性算法。证明了非平凡群与包含$Z^2$的非平凡交换群的限制圈积中的测地线问题是NP-困难的，并且存在一个多项式时间逼近方案.证明了两个非平凡阿贝尔群的限制圈积上的测地线问题是NP-难的当且仅当第二个阿贝尔群包含$Z^2$ .

Infinitely presented stable groups and invariant random subgroups of metabelian groups

We prove that all invariant random subgroups of the lamplighter group $L$ are co-sofic. It follows that $L$ is permutation stable, providing an example of an infinitely presented such a group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Quantifying conjugacy separability in wreath products of groups

We study generalisations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of $\mathcal{C}$-cyclic subgroup separability of $B$, and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are …

群的圈积中的方程

Equations in wreath products

该研究综述群圈积中方程的可解性，证明二次丢番图问题在阿贝尔群圈积及Baumslag有限表现亚阿贝尔群中可解，同时探讨了换位子宽度的相关问题。

Growth of actions of solvable groups

Given a finitely generated group $G$, we are interested in common geometric properties of all graphs of faithful actions of $G$. In this article we focus on their growth. We say that a group $G$ has a Schreier growth gap $f(n)$ if every faithful $G$-set $X$ satisfies $\mathrm{vol}_{G, X}(n)\succcurlyeq f(n)$, where $\mathrm{vol}_{G, X}(n)$ is the growth of the action of $G$ on $X$. Here we study Schreier growth gaps for finitely generated solvable groups. We prove that if a metabelian group $G$ is either finitely presented or torsion-free, then $G$ has a Schreier growth gap $n^2$, provided $G$…