On the residual nilpotence of pure Artin groups

Residual nilpotence and ordering in one-relator groups and knot groups

Let $G=< x,t\mid w>$ be a one-relator group, where $w$ is a word in $x,t$. If $w$ is a product of conjugates of $x$ then, associated with $w$, there is a polynomial $A_w(X)$ over the integers, which in the case when $G$ is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on $w$, that if all roots of $A_w(X)$ are real and positive then $G$ is bi-orderable, and that if $G$ is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the resu…

Residual Torsion-Free Nilpotence, Bi-Orderability and Two-Bridge Links

Residual torsion-free nilpotence has proven to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of parafree groups. This paper proves Mayland's assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$. We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link gro…

Residual properties of groups defined by basic commutators

In this paper we study the residual nilpotence of groups defined by basic commutators. We prove that the so-called Hydra groups as well as certain of their generalizations and quotients are, in the main, residually torsion-free nilpotent. By way of contrast we give an example of a group defined by two basic commutators which is not residually torsion-free nilpotent.

$A_{2,3,2n}$ 三角形 Artin 群的剩余有限性

Residual Finiteness of $A_{2,3,2n}$ Triangle Artin Groups

通过将 $A_{2,3,2n}$ 型三角形 Artin 群分解为群图，并证明每个群图相对于其顶点群具有有限高度，证明了所有 $n\geq4$ 时这些群是剩余有限的。

Note on residual finiteness of Artin groups

Let $A$ be an Artin group. A partition $\mathcal{P}$ of the set of standard generators of $A$ is called admissible if, for all $X,Y \in \mathcal{P}$, $X \neq Y$, there is at most one pair $(s,t) \in X \times Y$ which has a relation. An admissible partition $\mathcal{P}$ determines a quotient Coxeter graph $Γ/\mathcal{P}$. We prove that, if $Γ/\mathcal{P}$ is either a forest or an even triangle free Coxeter graph and $A_X$ is residually finite for all $X \in \mathcal{P}$, then $A$ is residually finite.

Conjugacy p-separability of right-angled Artin groups and applications

We prove that every subnormal subgroup of p-power index in a right-angled Artin group is conjugacy p-separable. As an application, we prove that every right-angled Artin group is conjugacy separable in the class of torsion-free nilpotent groups. As another application, we prove that the outer automorphism group of a right-angled Artin group is virtually residually p-finite. We also prove that the Torelli group of a right-angled group is residually torsion-free nilpotent, hence residually p-finite and bi-orderable.

Poly-freeness of even Artin groups of FC type

We prove that even Artin groups of FC type are poly-free and residually finite.

关于某些Artin群、可构造可解群、广义Baumslag-Solitar群及$SL_n(\Z)$的非交换外平方的剩余有限性

On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and $SL_n(\Z)$

本文证明多种群类（偶数Artin群、RAAG、极限群、Coxeter群、可解群、广义Baumslag-Solitar群及$SL_n(\Z)$等）的非交换外平方的剩余有限性，并建立上同调与同调良性的等价性。

Residual finiteness of certain 2-dimensional Artin groups

We show that many $2$-dimensional Artin groups are residually finite. This includes $3$-generator Artin groups with labels $\geq 4$ except for $(2m+1, 4,4)$ for any $m\geq 2$. As a first step towards residual finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed.

Splittings of triangle Artin groups

We show that a triangle Artin group $\text{Art}_{MNP}$ where $M\leq N\leq P$ splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either $M>2$, or $N>3$. We also prove that all even three generator Artin groups are residually finite.

正特征下Artin核的同调增长

Homological growth of Artin kernels in positive characteristic

该研究针对含直角Artin群、Bestvina--Brady群的一类剩余有限有理可解群，证明正特征下Lück逼近定理的类似结论，明确模p同调增长与泛可除环系数群同调维数的关系及剩余链无关性，给出一般RFRS群的不变量不等式，并拓展相关应用。

Hereditary conjugacy separability of right angled Artin groups and its applications

We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are conjugacy separable and have solvable conjugacy problem.

短层次超几何群II：商和Artin群的Hopf性质

Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups

本文证明大多数大类型和超几何类型的Artin群是Hopf群，并展示大部分五孔球面映射群的商是层次超几何的。

Artin群的有限 stature

Finite Stature in Artin groups

本文研究了 Artin 群的有限 stature 性质，通过图群分解方法推导出残余有限性，扩展了已知残余有限的 Artin 群范围。

无限群中的概率 nilpotence

Probabilistic nilpotence in infinite groups

本文研究了无限群的 k 步 nilpotence 度，证明了 finitely 生成群的 nilpotence 度与 virtually k 步 nilpotent 性的关系，并推广了有限群和有限生成群的结果。

Limit groups over coherent right-angled Artin groups

A new class of groups $\mathcal{C}$, containing all coherent RAAGs and all toral relatively hyperbolic groups, is defined. It is shown that, for a group $G$ in the class $\mathcal{C}$, the $\mathbb{Z}[t]$-exponential group $G^{\mathbb{Z}[t]}$ may be constructed as an iterated centraliser extension. Using this fact, it is proved that $G^{\mathbb{Z}[t]}$ is fully residually $G$ (i.e. it has the same universal theory as $G$) and so its finitely generated subgroups are limit groups over $G$. If $\mathbb{G}$ is a coherent RAAG, then the converse also holds - any limit group over $\mathbb{G}$ embeds…

On some of the residual properties of finitely generated nilpotent groups

In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol's work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS, a result which may be of independent interest though not directly applicable to 3-manifold topology. As a corollary, we deduce that any RFRS group cannot contain a nonabelian torsion-free nilpotent group. This result a…

Right-angled Artin subgroups of the $C^{\infty}$ diffeomorphism group of the real line

We prove that every right-angled Artin group embeds into the $C^{\infty}$ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the $C^{\infty}$ diffeomorphism group of the real line.

On subgroups of right angled Artin groups with few generators

For each natural number $d$ we construct a $3$-generated group $H_d$, which is a subdirect product of free groups, such that the cohomological dimension of $H_d$ is $d$. Given a group $F$ and a normal subgroup $N \lhd F$ we prove that any right angled Artin group containing the special HNN-extension of $F$ with respect to $N$ must also contain $F/N$. We apply this to construct, for every $d \in \mathbb{N}$, a $4$-generated group $G_d$, embeddable into a right angled Artin group, such that the cohomological dimension of $G_d$ is $2$ but the cohomological dimension of any right angled Artin grou…

Pure braid groups are not residually free

We show that the Artin pure braid group on at least four strands is not residually free. Our results also show that the pure braid group on at least three strands has corank two.

Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

Let G be a real semisimple Lie group with no compact factors and finite centre, and let $Λ$ be a lattice in G. Suppose that there exists a homomorphism from $Λ$ to the outer automorphism group of a right-angled Artin group $A_Γ$ with infinite image. We give an upper bound to the real rank of G that is determined by the structure of cliques in $Γ$. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup $\mathcal{T}}(A_Γ)$ of $Aut(A_Γ)$. We answer a question of Day relating to the abelianisation of $\mathcal{T}}(A_Γ)$, and show that $\mathcal{T}}(A_Γ)$ and its image in $…

Sur les representations de Krammer generiques

We define a representation of the Artin groups of type ADE by monodromy of generalized KZ-systems which is shown to be isomorphic to the generalized Krammer representation originally defined by A.M. Cohen and D. Wales, and independantly by F. Digne. It follows that all pure Artin groups of spherical type are residually torsion-free nilpotent, hence (bi-)orderable. Using that construction we show that these irreducible representations are Zariski-dense in the corresponding general linear group. It follows that all irreducible Artin groups of spherical type can be embedded as Zariski-dense subgr…

二维 Shephard 群

2-dimensional Shephard groups

研究二维 Shephard 群的非正曲率性质：证明大幂商非 CAT(0)，构造余紧 CAT(0) 分片欧氏胞腔复形，证明其 acylindrically hyperbolic 与 relatively hyperbolic，并由此得到一类二维 Artin 群的剩余有限性。

纯虚拟孪生群的结构与自同构

Structure and automorphisms of pure virtual twin groups

本文研究纯虚拟孪生群的结构与自同构，证明其为具有平凡中心的不可约直角 Artin 群，给出精确展示及无限秩自由群迭代半直积分解，完整刻画其自同构群，并由此推出虚拟孪生群剩余有限及纯虚拟孪生群具有 $R_\infty$-性质。