Bestvina-Brady Groups Research Articles

On the structure of finitely presented Bestvina–Brady groups

Right-angled Artin groups and their subgroups are of great interest because of their geometric, combinatorial and algorithmic properties. It is convenient to define these groups using finite simplicial graphs. The isomorphism type of the group is uniquely determined by the graph. Moreover, many structural properties of right-angled Artin groups can be expressed in terms of their defining graph. In this paper, we address the question of understanding the structure of a class of subgroups of right-angled Artin groups in terms of the graph. Bestvina and Brady, in their seminal work, studied these subgroups (now called Bestvina–Brady groups or Artin kernels) from a finiteness conditions viewpoint. Unlike the right-angled Artin groups the isomorphism type of Bestvina–Brady groups is not uniquely determined by the defining graph. We prove that certain finitely presented Bestvina–Brady groups can be expressed as an iterated amalgamated product. Moreover, we show that this amalgamated product can be read off from the graph defining the ambient right-angled Artin group.

Homological growth of Artin kernels in positive characteristic

We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina–Brady groups. Specifically, we prove that the mod p homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.

Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups

Abstract We give a characterization of finitely presented Bestvina–Brady groups which split over abelian subgroups and describe the JSJ-decompositions of those Bestvina–Brady groups.

On the virtual and residual properties of a generalization of Bestvina-Brady groups

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.

Virtually special non-finitely presented groups via linear characters

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.

On the Bieri–Neumann–Strebel–Renz Σ-invariants of the Bestvina–Brady groups

Abstract We study the Bieri–Neumann–Strebel–Renz invariants and we prove the following criterion: for groups H and K of type F ⁢ P n {FP_{n}} such that [ H , H ] ⊆ K ⊆ H {[H,H]\subseteq K\subseteq H} and a character χ : K → ℝ {\chi:K\to\mathbb{R}} with χ ⁢ ( [ H , H ] ) = 0 {\chi([H,H])=0} , we have [ χ ] ∈ Σ n ⁢ ( K , ℤ ) {[\chi]\in\Sigma^{n}(K,\mathbb{Z})} if and only if [ μ ] ∈ Σ n ⁢ ( H , ℤ ) {[\mu]\in\Sigma^{n}(H,\mathbb{Z})} for every character μ : H → ℝ {\mu:H\to\mathbb{R}} that extends χ. The same holds for the homotopical invariants Σ n ⁢ ( - ) {\Sigma^{n}(-)} when K and H are groups of type F n {F_{n}} . We use these criteria to complete the description of the Σ-invariants of the Bieri–Stallings groups G m {G_{m}} , and more generally to describe the Σ-invariants of the Bestvina–Brady groups. We also show that the “only if” direction of the above criterion holds if we assume only that K is a subnormal subgroup of H, where both groups are of type F ⁢ P n {FP_{n}} . We apply this last result to wreath products.

Identifying Dehn functions of Bestvina–Brady groups from their defining graphs

Let $$\Gamma $$ be a finite simplicial graph such that the flag complex on $$\Gamma $$ 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 $$\Gamma $$ .

Discriminantal bundles, arrangement groups, and subdirect products of free groups

We construct bundles $$E_k({{\mathscr {A}}},{{\mathscr {F}}}) \rightarrow M$$ over the complement M of a complex hyperplane arrangement $${{\mathscr {A}}}$$ , depending on an integer $$k \geqslant 1$$ and a set $${{\mathscr {F}}}=\{f_1, \ldots , f_\mu \}$$ of continuous functions $$f_i :M \rightarrow {{\mathbb {C}}}$$ whose differences are nonzero on M, generalizing the configuration space bundles arising in the Lawrence–Krammer–Bigelow representation of the pure braid group. We display such families $${{\mathscr {F}}}$$ for rank two arrangements, reflection arrangements of types $$A_\ell $$ , $$B_\ell $$ , $$D_\ell $$ , $$F_4$$ , and for arrangements supporting multinet structures with three classes, with the resulting bundles having nontrivial monodromy around each hyperplane. The construction extends to arbitrary arrangements by pulling back these bundles along products of inclusions arising from subarrangements of these types. We then consider the faithfulness of the resulting representations of the arrangement group $$\pi _1(M)$$ . We describe the kernel of the product $$\rho _{{\mathscr {X}}}:G \rightarrow \prod _{S \in {{\mathscr {X}}}} G_S$$ of homomorphisms of a finitely-generated group G onto quotient groups $$G_S$$ determined by a family $${{\mathscr {X}}}$$ of subsets of a fixed set of generators of G, extending a result of Theodore Stanford about Brunnian braids. When the projections $$G \rightarrow G_S$$ split in a compatible way, we show the image of $$\rho _{{\mathscr {X}}}$$ is normal with free abelian quotient, and identify the cohomological finiteness type of G. These results apply to some well-studied arrangements, implying several qualitative and residual properties of $$\pi _1(M)$$ , including an alternate proof of a result of Artal, Cogolludo, and Matei on arrangement groups and Bestvina–Brady groups, and a dichotomy for a decomposable arrangement $${{\mathscr {A}}}$$ : either $$\pi _1(M)$$ has a conjugation-free presentation or it is not residually nilpotent.

Abelian duality and propagation of resonance

We explore the relationship between a certain "abelian duality" property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the "EPY property." The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups, and Bestvina-Brady groups. Our approach brings to the fore the relevance of the Cohen-Macaulay condition in this combinatorial context.

The Dehn functions of Stallings–Bieri groups

We show that the Stallings–Bieri groups, along with certain other Bestvina–Brady groups, have quadratic Dehn function.

Arrangements of hypersurfaces and Bestvina–Brady groups

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 niteness properties.

Homological and homotopical Dehn functions are different

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, whereas the homotopical Dehn function measures fillings of curves by disks. Because the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions; however, before this work, there were no known examples of finitely presented groups for which the two functions differ. This paper gives such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina–Brady groups.

Pushing fillings in right-angled Artin groups

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 pushing along the height gradient, we also show that the k-th order Dehn function of a Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs called "orthoplex groups" which show that each of these upper bounds is sharp.

Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products

We compute: In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.

Hereditary conjugacy separability of right-angled Artin groups and its applications

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 cohomology of Bestvina–Brady groups

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 of the groups introduced by M. Bestvina and N. Brady in [3]. We compute the cohomological dimension of each of these groups.

An isoperimetric function for Bestvina-Brady groups

Given a right-angled Artin group A, the associated Bestvina–Brady group is defined to be the kernel of the homomorphism A → ℤ that maps each generator in the standard presentation of A to a fixed generator of ℤ. We prove that the Dehn function of an arbitrary finitely presented Bestvina–Brady group is bounded above by n4. This is the best possible universal upper bound.

Algebraic invariants for Bestvina-Brady groups

Bestvina–Brady groups arise as kernels of length homomorphisms GΓ → ℤ from right-angled Artin groups to the integers. Under some connectivity assumptions on the flag complex ΔΓ, we compute several algebraic invariants of such a group NΓ, directly from the underlying graph Γ. As an application, we give examples of finitely presented Bestvina–Brady groups which are not isomorphic to any Artin group or arrangement group.

Quasi-Kähler Bestvina-Brady groups

A finite simple graph Γ determines a right-angled Artin group GΓ, 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 NΓ is the kernel of the projection GΓ → Z, which sends each generator v to 1. We establish precisely which graphs Γ give rise to quasi-Kahler (respectively, Kahler) groups NΓ. This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasiprojective variety.

Some groups of type VF

A group is of type VF if it contains a finite-index subgroup which has a finite classifying space. We construct groups of type VF in which the centralizers of some elements of finite order are not of type VF and groups of type VF containing infinitely many conjugacy classes of finite subgroups. From these examples it follows that a group G of type VF need not admit a finite-type or finite classifying space for proper actions (sometimes also called the universal proper G-space). We construct groups G for which the minimal dimension of a universal proper G-space is strictly greater than the virtual cohomological dimension of G. Each of our groups embeds in some general linear group over the rational integers. Applications to algebraic K-theory of group algebras and topological K-theory of group C*-algebras are also considered. The groups are constructed as finite extensions of Bestvina-Brady groups.