S K Roushon
In this short note we prove that a class of Artin groups of affine and complex types are virtually poly-free, answering partially the question if all Artin groups are virtually poly-free.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.
We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-dimensional type. A particular case is that for $n \geq 4$, the $n$-strand braid group is not virtually cocompactly cubulated.
Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$.
The affine Artin group of type $\widetilde B_n$ is virtually poly-free
本研究证明$\tilde B_n$型仿射Artin群具有几乎多自由性质，且该证明过程为$\tilde B_n$的$K(π,1)$问题提供了新的解决思路。
We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group $A$ to be (virtually) cocompactly cubulated, in terms of the defining graph of $A$.
We study the algebraic structure of the automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved by the authors and John Crisp (arXiv:math/0610980) for two-dimensional right-angled Artin groups.
Virtual splittings of right-angled Artin groups
本文研究了右偏角Artin群是否能以秩为n的阿贝尔子群虚拟分裂的条件，证明了四个等价断言，揭示了群结构与图性质之间的深刻联系。
On the virtually cyclic dimension of normally poly-free groups
本文给出了正规多自由群虚拟循环维数关于长度的上界，并由此证明FC型虚拟偶Artin群存在有限维分类空间模型。
Balanced groups and the virtually cyclic dimension of poly-surfaces groups
本文通过研究平衡性质（Lück的条件C），获得正常多面体群和正常多自由群的虚拟循环维数的线性上界，证明了平衡性质在短正合列、极限和圆柱形图群分解下的稳定性，并应用该结果证明了正常多超几何群、正常多自由群和正常多面体群均为平衡群。
We prove that Artin groups from a class containing all large-type Artin groups are systolic. This provides a concise yet precise description of their geometry. Immediate consequences are new results concerning large-type Artin groups: biautomaticity; existence of $EZ$-boundaries; the Novikov conjecture; descriptions of finitely presented subgroups, of virtually solvable subgroups, and of centralizers for infinite order elements; the Burghelea conjecture; existence of low-dimensional models for classifying spaces for some families of subgroups.
For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.
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.
Property $R_\infty$ for Artin and Coxeter groups whose outer automorphisms are length-preserving
该研究证明非球面Artin与Coxeter群的保长自同构具无限Reidemeister数，进而确立多类Artin、Coxeter群的$R_\infty$性质，还证明无限不可约非仿射强刚性Coxeter群的自同构群具acylindrically双曲性。
Pure braid groups are RFRS
该研究证明纯辫群是RFRS，肯定了Agol2014年提出的辫群是否为虚拟RFRS的问题，还为Artin群是否虚拟特殊的问题提供了证据。
There exist right angled Artin groups $A$ such that the isomorphism problem for finitely presented subgroups of $A$ is unsolvable, and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows that if $S$ is a surface of finite type and the genus of $S$ is sufficiently large, then the corresponding decision problems for the mapping class group $Mod(S)$ are unsolvable. Every virtually special group embeds in the mapping class group of infinitely many closed surfaces. Examples are given of finitely presented subgroups of mapping class groups that h…
The Tits Alternative for two-dimensional Artin groups and Wise's Power Alternative
本文证明二维Artin群满足加强的Tits备选性质，并在相关Coxeter群双曲时，肯定回答Wise关于两个元素大幂生成子群的问题。
For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.
It is conjectured that the central quotient of every irreducible Artin group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin groups associated to triangle-free graphs and Artin groups of large type associated to cones over square-free bipartite graphs. In fact, we treat Artin groups that are known to be CAT(0) groups by a result of Brady and McCammond.
We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.
For a two-dimensional Artin group $A$ whose associated Coxeter group is hyperbolic, we prove that the action of $A$ on the hyperbolic space obtained by coning off certain subcomplexes of its modified Deligne complex is acylindrical. Moreover, if for each $s\in S$ there is $t\in S$ with $m_{st}< \infty$, then this action is universal. As a consequence, for $|S|\geq 3$, if $A$ is irreducible, then it is acylindrically hyperbolic. We also obtain the Tits alternative for $A$, and we classify the subgroups of $A$ that virtually split as a direct product. A key ingredient in our approach is a simple…
We prove that any large even Artin group is poly-free and that any even Artin group based on a triangle graph is also poly-free.
We prove that even Artin groups of FC type are poly-free and residually finite.
Lattice envelopes of right-angled Artin groups
本文构造非联图情形下直角 Artin 群的局部紧余紧格包络，并证明其半直积图自同构群为泛格包络，还刻画相关 Cayley 图全自同构群。
A Deligne complex for virtual Artin groups
研究虚拟阿廷群，通过构建德利涅复形类似物对其展开几何研究，证明对局部可约定义图是CAT(0)，并应用于有限子群分类及证明$K(\pi, 1)$猜想类似物，得出群几乎无挠且有最小维度余紧分类空间模型等结论。
For a group $G$ we consider the classifying space $E_{\mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{\mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of Lück-Reich-Rognes-Varisco for Artin groups. We then study the conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube g…
Given a discrete group $G$, for any integer $r\geqslant0$ we consider the family of all virtually abelian subgroups of $G$ of rank at most $r$. We give an upper bound for the Bredon cohomological dimension of $G$ for this family for a certain class of groups acting on $\mathrm{CAT}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of J.-F. Lafont.
We show that every right-angled Artin group AG defined by a graph G of finite chromatic number is poly-free with poly-free length bounded between the clique number and the chromatic number of G. Further, a characterization of all right-angled Artin groups of poly-free length 2 is given, namely the group AG has poly-free length 2 if and only if there exists an independent set of vertices D in G such that every cycle in G meets D at least twice. Finally, it is shown that AG is a semidirect product of 2 free groups of finite rank if and only if G is a finite tree or a finite complete bipartite gr…
A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.
On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group
本文研究了右角Artin群的$\ell^2$-Betti数及外自同构群的代数纤维化特性，揭示了其非零条件与虚拟纤维化的关系。
Automorphism and outer automorphism groups of Right-angled Artin groups are not relatively hyperbolic
本文证明定义图至少含3个顶点的右角Artin群的自同构群，以及非实质上同构于单顶点或不连通定义图的外自同构群，均不是相对双曲的。
In this article we construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the associated automatic subgroups, subject to further geometric conditions. These results are proved in the general context of fundamental groups of graphs of groups. The hypotheses of our closure results are satisfied in a variety of examples such as Artin groups of sufficiently large type, Coxeter groups, virtually abelian groups, and groups that are hyperbolic relati…
Let $G$ be a polycyclic, metabelian or soluble of type (FP)$_{\infty}$ group such that the class $Rat(G)$ of all rational subsets of $G$ is a boolean algebra. Then $G$ is virtually abelian. Every soluble biautomatic group is virtually abelian.
We give a condition on the defining graph of a right-angled Artin group which implies its automorphism group is virtually indicable, that is, it has a finite-index subgroup that admits a homomorphism onto $\Z$. We use this as part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.
A finitely generated group $Γ$ is called strongly scale-invariant if there exists an injective endomorphism $φ: Γ\to Γ$ with the image $φ(Γ)$ of finite index in $Γ$ and the subgroup $\displaystyle \bigcap_{n>0}φ^n(Γ)$ finite. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this paper, we study this question for the …
Splittings and poly-freeness of triangle Artin groups
该研究刻画了三角阿廷群可分裂为自由群图的充要条件，完全解决了相关公开问题，并部分证明了这类群的虚多自由性，推进了群论领域的相关研究。
We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.
On the Isoperimetric functions of a class of Artin Groups
本文针对定义图中不含标号为3的边的Artin群，利用相对扩展表示的small cancellation理论，构造出n^6多项式等周函数，从而证明偶Artin群的字问题可解。
A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(\mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy proble…
Profinite properties of algebraically clean graphs of free groups
本文证明了代数干净的群图对每个素数 $p$ 具有几乎剩余 $p$-有限和上同调 $p$-完备性，且上同调良好，并应用于二维 Artin 群。
Flat braid groups, right-angled Artin groups, and commensurability
本文研究了平坦辫群与直角Artin群的可公度性，证明了对于 $n=7$ 或 $\geq 11$ 时 $\mathrm{PFB}_n$ 并非实质上的直角Artin群，推翻了相关猜想，并发现 $\mathrm{FB}_7$ 与 $A(P_4)$ 可公度。
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。
We calculate the Bieri-Neumann-Strebel-Renz invariant $Σ^1(G)$ for even Artin groups $G$ with underlying graph $Γ$ such that if there is a closed reduced path in $Γ$ with all labels bigger than 2 then the length of such path is always odd. We show that $Σ^1(G)^c$ is a rationally defined spherical polyhedron.
We study the (virtual) indicability of the automorphism group $Aut(A_Γ)$ of the right-angled Artin group $A_Γ$ associated to a simplicial graph $Γ$. First, we identify two conditions -- denoted (B1) and (B2) -- on $Γ$ which together imply that $H^1(G, Z)=0$ for certain finite-index subgroups $G<Aut(A_Γ)$. On the other hand we will show that (B2) is equivalent to the matrix group ${\mathcal H} = {\rm Im}(Aut(A_Γ) \to Aut(H_1(A_Γ))) <GL(n,Z)$ not being virtually indicable, and also to $\mathcal H$ having Kazhdan's property (T). As a consequence, $Aut(A_Γ)$ virtually surjects onto $Z$ whenever $Γ…
We characterize the virtually nilpotent finitely generated groups (or, equivalently by Gromov's theorem, groups of polynomial growth) for which the Domino Problem is decidable: These are the virtually free groups, i.e. finite groups, and those having $\Z$ as a subgroup of finite index.
We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a right-angled Coxeter group.
On the linearity of fundamental groups of certain graphs of groups
本文研究顶点群为自由群、自由阿贝尔群或直角 Artin 群的玫瑰图与星图群图基本群，并在不同边群条件下证明其在 $\mathbb{Z}$ 上线性。
Configuration Lie groupoids and orbifold braid groups
本文提出配置李群胚的两种定义，证明一类李群胚的Fadell-Neuwirth型纤维化定理，并利用纯轨道折叠辫群的多重几乎自由结构，推导出相应Artin群及轨道折叠辫群的Farrell-Jones猜想。
Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equ…
We compute the $p$-central and exponent-$p$ series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to a partially commuting polynomial ring and power series ring. We finally show how the growth series of these various objects are related to each other.
Growth of automorphisms of virtually special groups
研究虚拟特殊群的外自同构增长速度，证明其增长为多项式或指数，并构建了类似曲面同胚的分解。
In this paper we generalize the notion of strongly poly-free group to a larger class of groups, we call them strongly poly-surface groups and prove that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for any virtually strongly poly-surface group. A consequence is that the Whitehead group of a torsion free subgroup of any virtually strongly poly-surface group vanishes.
In this note we point out an error in the above paper and refer to some papers where this error is corrected and a more general theorem is proved.
We investigate the Hurwitz action of the braid group Br_n on the n-fold Cartesian product of Br_3 and determine some stabilisers of its Artin systems. Our algebraic result is complemented by a geometric study of families of plane polynomial coverings of degree 3. Together they lead to characterisations of the set of 'paths realised by degenerations' of the polynomials as defined by Donaldson in 'Polynomials, vanishing cycles and Floer homology'.
We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the …
Taming Reidemeister zeta functions on virtually polycyclic groups
本文为几乎多循环群证明 Reidemeister 数公式，引入驯服 Reidemeister 数，证明其 zeta 函数有理，并由此推出普通 Reidemeister 及 Nielsen zeta 函数的有理性。
We provide two simple proofs of the fact that even Artin groups of FC-type are poly-free which was recently established by R. Blasco-Garcia, C. Martínez-Pérez and L. Paris. More generally, let $Γ$ be a finite simplicial graph with all edges labelled by positive even integers and $A_Γ$ be its associated Artin group; our new proof implies that if $A_T$ is poly-free (resp. normally poly-free) for every clique $T$ in $Γ$, then $A_Γ$ is poly-free (resp. normally poly-free). We prove similar results regarding the Farrell--Jones Conjecture for even Artin groups. In particular, we show that if $A_Γ$ i…
Liftable automorphisms of right-angled Artin groups
研究右角Artin群的可提升自同构及其相关同态性质
This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group $G$ has an element whose normal closure is abelian and of finite index, then $G$ has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).
We show that any soluble group $G$ of type Bredon-$\FP_{\infty}$ with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type $\FP_{\infty}$ must be virtually cyclic. To prove this, we first reduce the problem to the case of polycyclic groups and then we show that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-$\FP_n$ for some $n \leq 3$ and restrict to abelian-by-nilpotent, …
Congruence subgroups of small Artin and Coxeter groups
本文研究小Artin群与Coxeter群的同余子群，基于Squier矩阵表示在特定取值下的整性，推广辫群同余子群理论，并确定低层级主同余子群。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。