We investigate the relationship between the geometric Bieri-Neumann-Strebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as Kähler and quasi-Kähler manifolds.
We calculate the Bieri-Neumann-Strebel-Renz invariant $Σ^1(G)$ for even Artin groups $G$ with underlying graph $Γ$ such that if there is a closed reduced path in $Γ$ with all labels bigger than 2 then the length of such path is always odd. We show that $Σ^1(G)^c$ is a rationally defined spherical polyhedron.
For a group $G$ that is a limit group over Droms RAAGs such that $G$ has trivial center, we show that $Σ^1(G) = \emptyset = Σ^1(G, \mathbb{Q})$. For a group $H$ that is a finitely presented residually Droms RAAG we calculate $Σ^1(H)$ and $Σ^2(H)_{dis}$. In addition, we obtain a necessary condition for $[χ]$ to belong to $Σ^n(H)$.
For a finitely generated group $G$ we calculate the Bieri-Neumann-Strebel-Renz invariant $Σ^1(\X(G))$ for the weak commutativity construction $\X(G)$. Identifying $S(\X(G))$ with $S(\X(G) / W(G))$ we show $Σ^2(\X(G),\Z) \subseteq Σ^2(\X(G)/ W(G),\Z)$ and $Σ^2(\X(G)) \subseteq $ $ Σ^2(\X(G)/ W(G))$ that are equalities when $W(G)$ is finitely generated and we explicitly calculate $Σ^2(\X(G)/ W(G),\Z)$ and $ Σ^2(\X(G)/ W(G))$ in terms of the $Σ$-invariants of $G$. We calculate completely the $Σ$-invariants in dimensions 1 and 2 of the group $ν(G)$ and show that if $G$ is finitely generated group …
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…
We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.
Given a finite simplicial graph ${\cal G}$, the graph group $G{\cal G}$" is the group with generators in one-to-one correspondence with the vertices of ${\cal G}$ and with relations stating two generators commute if their associated vertices are adjacent in ${\cal G}$. The Bieri-Neumann-Strebel invariant can be explicitly described in terms of the original graph ${\cal G}$ and hence there is an explicit description of the distribution of finitely generated normal subgroups of $G{\cal G}$ with abelian quotient. We construct Eilenberg-MacLane spaces for graph groups and find partial extensions o…
Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing $G$ as the mapping torus of a free group automorphism. This is similar to the case for 3--manifold groups, and different from the case of mapping tori of exponentially growing free group automorphisms. The proof uses a hierarchical decomposition of $G$ and requires determining the Bieri-Neumann-Strebel invariant of th…
We investigate Friedl-Lück's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb{Z}$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the unive…
We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS)…
We show that several classes of groups G of PL-homeomorphisms of the real line admit non-trivial homomorphisms from G to the additive group of reals that are fixed by every automorphism of G. The classes of groups enjoying the stated property include the generalisations of Thompson's group F studied by K. S. Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.
We consider a class of metabelian groups first studied by Baumslag and Stammbach and we show that these groups are consistent with the Bieri-Groves conjecture which relates cohomological finiteness conditions to the Bieri-Neumann-Strebel sigma invariant.
Thompson's group F is the group of all increasing dyadic piecewise linear homeomorphisms of the closed unit interval. We compute Sigma^m(F) and Sigma^m(F;Z), the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F, and we show that Sigma^m(F) = Sigma^m(F;Z). As an application, we show that, for every m, F has subgroups of type F_{m-1} which are not of type F_{m}.
The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz invariants is proved over a field. Under certain hypotheses the Product Conjecture is shown to also hold over Z, even though D. Schuetz has recently shown that the Conjecture is false in general over Z. Our version over Z is applied in a joint paper with D. Kochloukova to derive new information about subgroups of Thompson's group F, namely that F has subgroups F_m which are not of type F_{m+1}.
We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones.
We use Bieri-Strebel invariants to determine when a normal fibre product in the product of two finitely presented groups is finitely presented. We give conditions that imply and in some cases characterize the existence of such finitely presented fibre products.
The observation that the 0-dimensional Geometric Invariant $Σ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincaré's limit set $Λ(Γ)$ of a discrete group $Γ$ of Möbius transformations (which contains the horospherical limit set of $Γ$) to the roots of tropical geometry (closely related to $Σ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $Σ(M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On …
The Sigma-invariants of Bieri-Neumann-Strebel and Bieri-Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Sigma-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The "0th stage" of this was developed in our paper [BG16]. The present pap…
A group $G$ is said to have the property $R_\infty$ if every automorphism $ϕ\in {\rm Aut}(G)$ has an infinite number of $ϕ$-twisted conjugacy classes. Recent work of Gonçalves and Kochloukova uses the $Σ^n$ (Bieri-Neumann-Strebel-Renz) invariants to show the $R_{\infty}$ property for a certain class of groups, including the generalized Thompson's groups $F_{n,0}$. In this paper, we make use of the $Ω^n$ invariants, analogous to $Σ^n$, to show $R_{\infty}$ for certain finitely generated groups. In particular, we give an alternate and simpler proof of the $R_{\infty}$ property for BS(1,n). Moreo…
We introduce a homology theory for subspace arrangements, and use it to extract a new system of numerical invariants from the Bieri-Neumann-Strebel invariant of a group. We use these to characterize when the set of basis conjugating outer automorphisms (a.k.a. the pure symmetric outer automorphism group) of a right-angled Artin group is itself a right-angled Artin group.
We describe some of the connections between the Bieri-Neumann-Strebel-Renz invariants, the Dwyer-Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, translated components in the characteristic varieties affect the answer. We illustrate this theory in the setting of toric complexes, as well as smooth, complex projective and quasi-projective varieties, with…
We compute the Bieri-Neumann-Strebel invariants $Σ^1$ for the generalized solvable Baumslag-Solitar groups $Γ_n$ and their finite index subgroups. Using $Σ^1$, we show that certain finite index subgroups of $Γ_n$ cannot be isomorphic to $Γ_{k}$ for any $k$. In addition, we use the BNS-invariants to give a new proof of property $R_\infty$ for the groups $Γ_n$ and their finite index subgroups.
In 2003 Bieri and Geoghegan generalized the Bieri-Neuman-Strebel invariant $Σ^1$ by defining $Σ^1(ρ)$, $ρ$ an isometric action by a finitely generated group $G$ on a proper CAT(0) space $M$. In this paper, we show how the natural and well-known connection between Bass-Serre theory and covering space theory provides a framework for the calculation of $Σ^1(ρ)$ when $ρ$ is a cocompact action by $G = B \rtimes A$, $A$ a finitely generated group, on a locally finite Bass-Serre tree $T$ for $A$. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endp…
We use Fox calculus to assign a marked polytope to a `nice' group presentation with two generators and one relator. Relating the marked vertices to Novikov-Sikorav homology we show that they determine the Bieri-Neumann-Strebel invariant of the group. Furthermore we show that in many cases the marked polytope is an invariant of the underlying group and that in those cases the marked polytope also determines the minimal complexity of all the associated HNN-splittings.
We present a full description of the Bieri-Neumann-Strebel invariant of restricted permutational wreath products of groups. We also give partial results about the 2-dimensional homotopical invariant of such groups. These results may be turned into a full picture of these invariants when the abelianization of the basis group is infinite. We apply these descriptions to the study of the Reidemeister number of automorphisms of wreath products in some specific cases.
We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope for finite free $G$-CW complexes together with a fixed choice of a ring homomorphism from the group ring $\mathbb{Z} G$ to a skew field. For the particular choice of the Linnell skew field $\mathcal{D}(G)$, this approach recovers most of the information encoded in the corresponding $L^2$-invariants. As an application, we prove that for agrarian groups of deficiency $1$, t…
The Bieri-Neumann-Strebel-Renz invariants $Σ^q(X,\mathbb{Z})\subset H^1(X,\mathbb{R})$ of a connected, finite-type CW-complex $X$ are the vanishing loci for Novikov-Sikorav homology in degrees up to $q$, while the characteristic varieties $\mathcal{V}^q(X) \subset H^1(X,\mathbb{C}^{\times})$ are the nonvanishing loci for homology with coefficients in rank 1 local systems in degree $q$. We show that each BNSR invariant $Σ^q(X,\mathbb{Z})$ is contained in the complement of the tropical variety associated to the algebraic variety $\mathcal{V}^{\le q}(X)$, and provide applications to several class…
We prove that if a right-angled Artin group $A_Γ$ is abstractly commensurable to a group splitting non-trivially as an amalgam or HNN-extension over $\mathbb{Z}^n$, then $A_Γ$ must itself split non-trivially over $\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_Γ$ and $A_Δ$ are commensurable and $Γ$ has no separating $k$-cliques for any $k\le n$ then neither does $Δ$, so "smallest size of separating clique" is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for $n\ge 4$ the…
The Stein group $F_{2,3}$ is the group of orientation-preserving homeomorphisms of the unit interval with slopes of the form $2^p3^q$ ($p,q\in\mathbb{Z}$) and breakpoints in $\mathbb{Z}[\frac{1}{6}]$. This is a natural relative of Thompson's group $F$. In this paper we compute the Bieri-Neumann-Strebel-Renz (BNSR) invariants $Σ^m(F_{2,3})$ of the Stein group for all $m\in\mathbb{N}$. A consequence of our computation is that (as with $F$) every finitely presented normal subgroup of $F_{2,3}$ is of type $\textrm{F}_\infty$. Another, more surprising, consequence is that (unlike $F$) the kernel of…
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-…
This is the second of two papers but has been written so as to have minimal dependence on the first paper (which is also on this archive). Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Assume G is of type F_n (type F_1 is finitely generated, type F_2 is finitely presented etc.) The "boundary", bdM, of M at infinity has two customary topologies - the compact "cone" topology and the Tits metric topology. We associate with any isometric action of G on M two subsets of …
This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study "Infinite groups as geometric objects", as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favorably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various me…
