Generalized Grigorchuk's Overgroups as points on $\mathcal{M}_k$

Röver's Simple Group is of Type $F_\infty$

We prove that Claas Röver's Thompson-Grigorchuk simple group $V\mathcal{G}$ has type $F_\infty$. The proof involves constructing two complexes on which $V\mathcal{G}$ acts: a simplicial complex analogous to the Stein complex for $V$, and a polysimiplical complex analogous to the Farley complex for $V$. We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

Boolean lattices in finite alternating and symmetric groups

Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilizers of chains of regular partitions, and the other type arises by taking stabilizers of chains of regular product structures. As an application, we prove in th…

关于分支群有限生成子群的结构

On the structure of finitely generated subgroups of branch groups

本文描述了具有子群归纳性质的分支群（如第一Grigorchuk群和挠GGS群）有限生成子群的块结构。

Quantifying lawlessness in finitely generated groups

We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" $\mathcal{A}_Γ : \mathbb{N} \rightarrow \mathbb{N}$. We show that $\mathcal{A}_Γ$ is bounded iff $Γ$ has a nonabelian free subgroup. By contrast we construct, for any nondecreasing unbounded function $f: \mathbb{N} \rightarrow \mathbb{N}$, an elementary amenable lawless groups for which $\mathcal{A}_Γ$ grows more slowly that $f$. We produce torsion lawless groups for which $\mathcal{A}_Γ$ is at least linear using Golod-Shafarevich theory, and give some upper bounds on $…

$\boldsymbol{\textit{C}}$-遗传共轭可分群与 wreath 积

$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products

该研究给出限制 wreath 积为 $\boldsymbol{\textit{C}}$-遗传共轭可分群的充要条件，证明 Grigorchuk 群是 2-遗传共轭可分群，构造了无穷多类非多循环遗传共轭可分群。

$\mathcal C$-graph automatic groups

We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov (arXiv:1107.3645) by replacing the regular languages in their definition with more powerful language classes. For a fixed language class $\mathcal C$, we call the resulting groups $\mathcal C$-graph automatic. We prove that the class of $\mathcal C$-graph automatic groups is closed under change of generating set, direct and free product for certain classes $\mathcal C$. We show that for quasi-realtime counter-graph automatic groups where normal forms have length that is linear in the geod…

Densely related groups

We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at all scales. Here, new relations means relations that do not follow from relations of smaller size. Being densely related is a quasi-isometry invariant among finitely generated groups. We check that a densely related group has none of its asymptotic cones simply connected. In particular a lacunary hyperbolic group cannot be densely related. We prove that the G…

Growth Tight Actions

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of $G$ on $\mathcal{X}$ is a growth tight action. It follows that if $\mathcal{X}$ is a cocompact, relatively hyperbolic $G$--space, then the action of $G$ on $\mathcal{X}$ is a growth tight action. This g…