Deficiency and abelianized deficiency of some virtually free groups

Normal Subgroups of Profinite Groups of Non-negative Deficiency

We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup…

Pro-p groups of positive deficiency

Let Γbe a finitely presentable pro-p group with a nontrivial finitely generated closed normal subgroup N of infinite index. Then def(Γ)\leq 1, and if def(Γ)=1 then Γis a pro-p duality group of dimension 2, N is a free pro-p group and Γ/N is virtually free. In particular, if the centre of Γis nontrivial and def(Γ)\geq 1, then def(Γ)=1, cd G \leq 2 and Γis virtually a direct product F \times Z_p, with F a finitely generated free pro-p group.

Constructing presentations of subgroups of right-angled Artin groups

Let $G$ be the right-angled Artin group associated to the flag complex $Σ$ and let $π:G\to\Z$ be its canonical height function. We investigate the presentation theory of the groups $Γ_n=π^{-1}(n\Z)$ and construct an algorithm that, given $n$ and $Σ$, outputs a presentation of optimal deficiency on a minimal generating set, provided $Σ$ is triangle-free; the deficiency tends to infinity as $n\to\infty$ if and only if the corresponding Bestvina-Brady kernel $\bigcap_nΓ_n$ is not finitely presented, and the algorithm detects whether this is the case. We explain why there cannot exist an algorithm…

On $p$-deficiency in groups

Recently, Schlage-Puchta proved super multiplicity of $p$-deficiency for normal subgroups of $p$-power index. We extend this result to all normal subgroups of finite index. We then use the methods of the proof to show that some groups with non-positive $p$-deficiency have virtually positive $p$-deficiency. We also compute the $p$-deficiency in some cases such as Fuchsian groups and study related invariants: the lower and upper absolute $p$-homology gradients and the $p$-Euler characteristic.

关于自由循环群和亏度1群的子群可分性

On subgroup separability of free-by-cyclic and deficiency 1 groups

本文证明了多项式增长单演的自由循环群在特定条件下子群可分，并指出随机亏度1群大概率不可分。

凝聚RFRS群

Coherent RFRS groups

本文研究上同调维数至多2的有限生成虚拟RFRS群的凝聚性等价条件，据此刻画二维Coxeter群等类群的凝聚性并证实相关猜想，附录中解决了立方化情形下的相关公开问题。