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.
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.
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.
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 $Γ$.
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.
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 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…
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…
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.
We present a new method for showing that groups are virtually special. This is done by considering finite quotients and linear characters. We use this to show that an infinite family of groups, related to Bestvina-Brady groups and branching, provides new examples of virtually special groups outside of a hyperbolic context.
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.
It is well known that every word hyperbolic group has a well-defined visual boundary. An example of C. Croke and B. Kleiner shows that the same cannot be said for CAT(0) groups. All boundaries of a CAT(0) group are, however, shape equivalent, as observed by M. Bestvina and R. Geoghegan. Bestvina has asked if they also satisfy the stronger condition of being cell-like equivalent. This article describes a construction which will produce CAT(0) groups with multiple boundaries. These groups have very complicated boundaries in high dimensions. It is our hope that their study may provide insight int…
We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb Z$ with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb H^4$ with fundamental domain the $120$-cell or the $24$-cell.
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.
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…
M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is "Yes" in the situation where the group in question splits as a direct product with infinite factors. We accomplish this by proving an interesting theorem in shape theory.
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…
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…
We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3. Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with finite volume, of complex-dimension n > 1. The group G is acting on the universal cover of M, which is isometric to H^n_C. It …
We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.
In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT(0) groups which are thick of order n and with polynomial divergence of order n+1, both these phenomena are new. With respect to thickness, these examples show the non-triviality at each level of the thickness hierarchy defined by Behrstock-Drutu-Mosher. With respect to divergence our examples resolve questions of Gromov and Gersten (the divergence questions were also recently and independe…
We use the projection complex machinery of Bestvina--Bromberg--Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to trees, then the group is quasiisometric to a finite-dimensional CAT(0) cube complex. We deduce various properties, including the Helly property for hierarchically quasiconvex subsets.
Croke-Kleiner admissible groups firstly introduced by Croke-Kleiner belong to a particular class of graph of groups which generalize fundamental groups of $3$--dimensional graph manifolds. In this paper, we show that if $G$ is a Croke-Kleiner admissible group, acting geometrically on a CAT(0) space $X$, then a finitely generated subgroup of $G$ has finite height if and only if it is strongly quasi-convex. We also show that if $G \curvearrowright X$ is a flip CKA action then $G$ is quasi-isometric embedded into a finite product of quasi-trees. With further assumption on the vertex groups of the…
This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to relate…
We investigate strictly developable simple complexes of groups with arbitrary local groups, or equivalently, group actions admitting a strict fundamental domain. We introduce a new method for computing the cohomology of such groups. We also generalise Bestvina's construction to obtain a polyhedral complex equivariantly homotopy equivalent to the standard development of the lowest possible dimension. As applications, for a group acting chamber transitively on a building of type $(W,S)$, we show that its Bredon cohomological dimension is equal to the virtual cohomological dimension of $W$ and gi…
