Connectivity at infinity for right angled Artin groups

Abstract

Abstract. We establish sufficient conditions implying semistability and connectivity at infinity properties for CAT(0) cubical complexes. We use this, along with the geometry of cubical K(π, 1)'s to give a complete description of the higher connectivity at infinity properties of right angled Artin groups. Among other things, this determines which right angled Artin groups are duality groups. Applications to group extensions are also included.

Divergence, undistortion and Hölder continuous cocycle superrigidity for full shifts

“…is connected then A is always has polynomial divergence [3,Corollary 4.8]. Furthermore, if has at least three vertices then A has one end when is connected [7,Theorem B] or [34,Corollary 5.2]. Because A projects onto Z n , where n is the number of vertices in , by adding all commuting relations between the generators, for any generator s in A , the length of s k in A is greater than or equal to the length ofs k…”

An algebraic characterization of 𝑘–colorability

“…(1) The graph Γ is a nontrivial join if and only if A(Γ) decomposes as a nontrivial direct product [18]. 2The graph Γ is disconnected if and only if A(Γ) decomposes as a nontrivial free product [2]. 3The graph Γ is square-free if and only if A(Γ) does not contain a subgroup isomorphic to a product F 2 × F 2 of nonabelian free groups [11,12].…”

A note on virtual duality and automorphism groups of right-angled Artin groups

“…With hindsight, the failure of duality here should not be too surprising, simply because right-angled Artin groups themselves are rarely duality groups. A wonderful theorem of Brady and Meier [8] shows that a right-angled Artin group A Γ is a duality group if and only if the flag complex Γ of the defining graph is Cohen-Macaulay (see Definition 2.1). To briefly sketch how their result implies Theorem A, let ∆ be the join of the subgraphs Γ 1 and Γ 2 given in Figure 1.…”