Stefan Papadima, Alexander I. Suciu
Bestvina-Brady groups arise as kernels of length homomorphisms from right-angled Artin groups G_\G to the integers. Under some connectivity assumptions on the flag complex Δ_\G, we compute several algebraic invariants of such a group N_\G, directly from the underlying graph \G. As an application, we give examples of Bestvina-Brady groups which are not isomorphic to any Artin group or arrangement group.
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Given a right-angled Artin group A, the associated Bestvina-Brady group is defined to be the kernel of the homomorphism A \to \mathbb{Z} that maps each generator in the standard presentation of A to a fixed generator of \mathbb{Z}. We prove that the Dehn function of an arbitrary finitely presented Bestvina-Brady group is bounded above by n^4. This is the best possible universal upper bound.
Complete classification of the Dehn functions of Bestvina-Brady groups
本文证明了每个有限表示的Bestvina-Brady群的Dehn函数增长为线性、二次、三次或四次多项式，并给出了确定多项式次数的显式判据，进而识别了阻碍某些Bestvina-Brady群具有CAT(0)结构的障碍。
Abelian Splittings and JSJ-Decompositions of Bestvina--Brady Groups
本文刻画了在阿贝尔子群上分裂的Bestvina--Brady群，并给出了其JSJ分解的描述。
Let K be a 2-dimensional finite flag complex. We study the CAT(0) dimension of the `Bestvina-Brady group', or `Artin kernel', Gamma_K. We show that Gamma_K has CAT(0) dimension 3 unless K admits a piecewise Euclidean metric of non-positive curvature. We give an example to show that this implication cannot be reversed. Different choices of K lead to examples where the CAT(0) dimension is 3, and either (i) the geometric dimension is 2, or (ii) the cohomological dimension is 2 and the geometric dimension is not known.
Let $Γ$ be a finite simplicial graph such that the flag complex on $Γ$ is a $2$-dimensional triangulated disk. We show that with some assumptions, the Dehn function of the associated Bestvina--Brady group is either quadratic, cubic, or quartic. Furthermore, we can identify the Dehn function from the defining graph $Γ$.
On the structure of finitely presented Bestvina-Brady groups
本文针对Bestvina-Brady群同构型不由定义图唯一确定的问题，证明部分有限表现的该类群可表示为迭代融合积，且该结构可从外围直角Artin群的定义图中读取，丰富了相关群结构的图论刻画。
We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti numbers of Bestvina-Brady groups and of graph products of groups, * the weighted L^2-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.(However, for graph products of Coxeter groups this proviso is unnecessary.)
We compute the relative divergence and the subgroup distortion of Bestvina-Brady subgroups. We also show that for each integer $n\geq 3$, there is a free subgroup of rank $n$ of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. This result answers the question of Carr about the minimum rank $n$ such that some right-angled Artin group has a free subgroup of rank $n$ whose inclusion is not a quasi-isometric embedding. It is well-known that a right-angled Artin group $A_Γ$ is the fundamental group of a graph manifold whenever the defining graph $Γ$ is a tree. We sh…
For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus our computation gives the homology and cohomology of Bestvina-Brady groups. We compute the cohomological dimension of each of these groups over any field and over any subring of the rationals.
The Euler Characteristic of Spherical Bestvina-Brady Kernels
该研究计算所有球形型Artin群的Bestvina-Brady核的欧拉示性数，利用离散莫尔斯理论分析其分类空间结构，推广计算方法至满足特定条件的Artin群并给出实例。
We show that quasi-projective Bestvina-Brady groups are fundamental groups of complements to hyperplane arrangements. Furthermore we relate other normal subgroups of right-angled Artin groups to complements to arrangements of hypersurfaces. We thus obtain examples of hypersurface complements whose fundamental groups satisfy various finiteness properties.
A graphical description of the BNS-invariants of Bestvina-Brady groups and the RAAG recognition problem
本文通过图论描述有限表示Bestvina-Brady群的BNS不变量，证明其类包含RAAG类，并给出判定BBG是否为RAAG的准则。
A finite simple graph \G determines a right-angled Artin group G_\G, with one generator for each vertex v, and with one commutator relation vw=wv for each pair of vertices joined by an edge. The Bestvina-Brady group N_\G is the kernel of the projection G_\G \to \Z, which sends each generator v to 1. We establish precisely which graphs \G give rise to quasi-Kähler (respectively, Kähler) groups N_\G. This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasi-projective variety.
We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of $VR_t(S^n)$. Another application is to geometric gr…
Previously one of us introduced a family of groups $G^M_L(S)$, parametrized by a finite flag complex $L$, a regular covering $M$ of $L$, and a set $S$ of integers. We give conjectural descriptions of when $G^M_L(S)$ is either residually finite or virtually torsion-free. In the case that $M$ is a finite cover and $S$ is periodic, there is an extension with kernel $G_L^M(S)$ and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each other and prove many cases of them.
We study the Bieri-Neumann-Strebel-Renz invariants and we prove the following criterion: for groups $H$ and $K$ of type $FP_n$ such that $[H,H] \subseteq K \subseteq H$ and a character $χ: K \to \mathbb{R}$ with $χ([H,H]) = 0$ we have $[χ] \in Σ^n(K, \mathbb{Z})$ if and only if $[μ] \in Σ^n(H, \mathbb{Z})$ for every character $μ: H \to \mathbb{R}$ that extends $χ$. The same holds for the homotopical invariants $Σ^n(-)$ when $K$ and $H$ are groups of type $F_n$. We use these criteria to complete the description of the $Σ$-invariants of the Bieri-Stallings groups $G_m$ and more generally to desc…
Subgroups of Bestvina-Brady groups
本文刻画了Bestvina-Brady群的每个子群均为直角Artin群的图，并据此证实了这些群pro-p完备化的若干Galois理论猜想。
Matroids and isomorphism problems for Bestvina-Brady groups
本文通过拟阵理论分解Bestvina-Brady群的图同构问题，证明有限表示仅依赖循环拟阵，并构造高连通图使拟阵不同但群同构，且对偶弦图定义的BBG为右角Artin群。
In this thesis we investigate the Dehn functions of two different classes of groups: subdirect products, in particular subdirect products of limit groups; and Bestvina-Brady groups. Let D = Γ_1 \times ... \times Γ_n be a direct product of n \geq 3 finitely presented groups and let H be a subgroup of D. Suppose that each Γ_i contains a finite index subgroup Γ_i' \leq Γ_i such that the commutator subgroup [D', D'] of D' = Γ_1' \times ... \times Γ_n' is contained in H. Suppose furthermore that, for each i, the subgroup Γ_i H has finite index in D. We prove that H is finitely presented and satisfi…
We give a combinatorial criterion for determining which Bestvina--Brady group is isomorphic to a right-angled Artin group.
Brady proved that there are hyperbolic groups with finitely presented subgroups that are not of type $FP_3$ (and hence not hyperbolic). We reprove Brady's theorem by presenting a new construction. Our construction uses Bestvina-Brady Morse theory, but does not involve branched coverings.
Automorphisms of Bestvina-Brady Groups: IA Rigidity, Arithmetic Commensurability, and Finiteness
本文研究了Bestvina-Brady群的自同构性质，证明了IA刚性、算术共轭性和有限性相关结论，并探讨了其外自同构群的有限性。
We show that the Stallings-Bieri groups, along with certain other Bestvina-Brady groups, have quadratic Dehn function.
We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.
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.
Coarse embeddings of products of trees as quasi-isometry invariants
研究茂密树乘积能粗嵌入到多种群中的最大因子数，利用此区分相关群并排除粗嵌入，基于树乘积到分层双曲空间的粗嵌入一般陈述得出这些应用。
The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to $\mathbb{Z}$ whose kernel has strong topological finiteness proper…
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。
In this paper we give a complete description of the Bieri-Neumann-Strebel-Renz invariants of the Lodha-Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha-Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$, answering question 112 from Oberwolfach Rep., 15(2):1579-…
For a connected graph L, let G(L) be a group with generators the vertex set of L, subject only to the relations that the ends of each edge commute. Now let H(L) be the kernel of the homomorphism from G(L) to the integers that takes each vertex to 1. M. Bestvina and N. Brady have shown that finiteness properties of H(L) are intimately related to the topology of the clique complex of L. We give a presentation for H(L), with generators the edges of L, and an infinite family of relators for each 1-cycle in L. In the case when the clique complex for L is simply-connected, we give a finite presentat…
We construct "pushing maps" on the cube complexes that model right-angled Artin groups (RAAGs) in order to study filling problems in certain subsets of these cube complexes. We use radial pushing to obtain upper bounds on higher divergence functions, finding that the k-dimensional divergence of a RAAG is bounded by r^{2k+2}. These divergence functions, previously defined for Hadamard manifolds to measure isoperimetric properties "at infinity," are defined here as a family of quasi-isometry invariants of groups; thus, these results give new information about the QI classification of RAAGs. By p…
Presentation of kernels of rational characters of right-angled Artin groups
本文刻画直角 Artin 群有理特征标核的有限生成与有限表示条件，给出有限生成集及表示，推广并代数证明了相关经典结果。
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 the $Σ$-invariants of Artin groups satisfying the $K(π,1)$-conjecture
本文研究了满足 $K(π,1)$ 猜想的 Artin 群的 $Σ$-不变量，扩展了特征属于该不变量的充分条件及部分逆命题，并以此推广了 Bestvina-Brady 定理的同调版本，完成了对不可约球面、仿射及三角形 Artin 群导出子群有限性质的分类。
For every atoroidal iwip automorphism $ϕ$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(ϕ)$ is obtained as "diagonal closure" of the support of the backward limit current $μ_-(ϕ)$. This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of $T_+(ϕ)$, in terms of Bestvina-Fe…
Generalisable presentations and compactness properties of locally compact right-angled Artin groups
本文提出可推广表示的概念，引入拓扑右角Artin群，研究其几何与紧性性质，并构造了具有特定有限性性质的离散群例子。
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