Topology of Random Right Angled Artin Groups

On the virtually-cyclic dimension of surface braid groups and right-angled Artin groups

We give a bound for the virtually cyclic dimension of groups with a normal subgroup of finite index which satisfies that every infinite virtually-cyclic subgroup is contained in a unique maximal such subgroup. As an application we provide a bound for the virtually-cyclic dimension for the braid group of a closed surface with genus greater than 2 and for right-angled Artin groups.

The depth of a Riemann surface and of a right-angled Artin group

We consider two families of spaces, $X$ : the closed orientable Riemann surfaces of genus $g>0$ and the classifying spaces of right-angled Artin groups. In both cases we compare the depth of the fundamental group with the depth of an associated Lie algebra, $L$, that can be determined by the minimal Sullivan algebra. For these spaces we prove that $$ \mbox{depth} \,\mathbb Q[π_1(X)] = \mbox{depth}\, {L}\,$$ and give precise formulas for the depth.

关于直角 Artin 群的 Gromov 猜想

On Gromov's conjecture for right-angled Artin groups

针对基本群为直角 Artin 群的 n 维闭 spin PSC 流形，本文证明其万有覆盖的宏观维度不超过 n−2，从而验证该类群情形下的 Gromov 猜想。

Topological 4-manifolds with right-angled Artin fundamental groups

We classify closed, topological spin$^+$ 4-manifolds with fundamental group $π$ of cohomological dimension $\leq 3$ (up to s-cobordism), after stabilization by connected sum with at most $b_3(π)$ copies of $S^2\times S^2$. In general we must also assume that $π$ also satisfies certain K-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of 3-manifolds and all right-angled Artin groups whose defining graphs have no 4-cliques.

直角 Artin 群的最小亏格问题

The minimal genus problem for right angled Artin groups

本文研究 RAAG 二阶同调类的最小亏格，给出基于卡积矩阵秩的下界，证明其在多类图中由环面不交并实现，并刻画单环面可表示类及反例。

Expanders and right-angled Artin groups

The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{Γ_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups $\{A(Γ_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of \emph{vector space expanders}, a novel struc…

Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups

The clique number of a random graph in the Erdos-Renyi model G(n,p) yields a random variable which is known to be asymptotically (as n tends to infinity) almost surely within one of an explicit logarithmic (on n) function r(n,p). We extend this fact by showing that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint complete subgraphs with as many vertices as r(n,p). The result is motivated by and applied to the sequential motion planning problem on random right angled Artin groups. Indeed, we give an asymptotical description of all the higher topological compl…

Artin经典辫形群中右角关系的组合起源

A combinatorial genesis of the right-angled relations in Artin's classical braid groups

本文通过组合方法研究Artin经典辫形群的右角关系，利用UC(n,p×q)空间的特性，推导出其基本群与Artin群的近似关系，并计算了相关空间的范畴和拓扑复杂度。

Right-angled Artin groups, polyhedral products and the TC-generating function

For a graph $Γ$, let $K(H_Γ,1)$ denote the Eilenberg-Mac Lane space associated to the right-angled Artin (RAA) group $H_Γ$ defined by $Γ$. We use the relationship between the combinatorics of $Γ$ and the topological complexity of $K(H_Γ,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal{O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fac…

超平面复形的同伦类型与右角Artin群的应用

Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups

研究通过分析准中位图中超平面复形的同伦类型，证明了右角Artin群的准等距性与特定复形的同伦等价性。

Homological stability for automorphism groups of RAAGs

We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.