arXiv subjects

Robert Bieri

Publications and source records attributed to Robert Bieri.

12 recordsLinked to original sources

arXiv2026

Piecewise isometry groups of Euclidean tessellations

Given a tessellation of Euclidean or hyperbolic space, the piecewise isometry group is the group whose elements are given by cutting space into finitely many tessellated convex subsets and gluing them back together. Groups of piecewise isometries of tessellations generalize Houghton's groups and Thompson's group $V$, and for cubical tessellations were studied by Bieri and Sach. We prove structure results about groups of piecewise isometries of sufficiently nice tessellations of Euclidean space, such as tessellations associated to crystallographic root systems, in particular proving that they are elementary amenable. Future work in progress will prove finite generation and higher finiteness properties.

math.GR↗

arXiv2021

Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations

Through the glasses of didactic reduction: We consider a (periodic) tessellation $\Delta$ of either Euclidean or hyperbolic $n$-space $M$. By a piecewise isometric rearrangement of $\Delta$ we mean the process of cutting $M$ along corank-1 tile-faces into finitely many convex polyhedral pieces, and rearranging the pieces to a new tight covering of the tessellation $\Delta$. Such a rearrangement defines a permutation of the (centers of the) tiles of $\Delta$, and we are interested in the group $PI(\Delta)$ of all piecewise isometric rearrangements of $\Delta$. In this paper we offer: a) An illustration of piecewise isometric rearrangements in the visually attractive hyperbolic plane, b) an explanation how this is related to Richard Thompson's groups, c) a chapter on the structure of the group pei$(\mathbb Z^n)$ of all piecewise Euclidean rearrangements of the standard tessellation of $\mathbb R^n$ by unit-cubes, and d) results on the finiteness properties of some subgroups of pei$(\mathbb Z^n)$.

math.GR↗

arXiv2018

Higher horospherical limit sets for G-modules over CAT(0) spaces

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 paper provides a higher-dimensional extension of the theory to the "nth stage" for any n.

math.GR↗

arXiv2016

Limit sets for modules over groups on CAT(0) spaces -- from the Euclidean to the hyperbolic

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 way we meet instances where $Σ(M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.

math.GR↗

arXiv2016

On Groups of PL-homeomorphisms of the Real Line

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G are studied which depend on three parameters I, A, and P and which generalize the local definition of Thompson's group F thus: G consists of all orientation preserving PL-homeomorphisms of the real line with supports in the interval I, slopes in the multiplicative subgroup P of the positive reals and breaks in a finite subset of the additive P submodule A of R. A first aim of the monograph is to investigate in which form familiar properties of F continue to hold for these groups. Main aims of the monograph are the determination of isomorphisms among the groups G and the study of their automorphism groups. Complete answers are obtained if the group P is not cyclic or if the interval I is the full line.

math.GR↗

arXiv2016

Groups of piecewise isometric permutations of lattice points

Let M denote either Euclidean or hyperbolic n-space, and let G be a discrete group of isometries of M, with the property that G respects and acts tile-transitively on a convex-polyhedral tesselation of M. Given an arbitrary base point p in M, we consider the orbit Gp in M and define a notion of "G-polyhedral pieces" S in Gp. The objects of our interest are the groups pi(S) of all piecewise G-isometric permutations on S. In this paper we merely present the two most basic examples, and these play rather different roles: The case when G = PSL(2,Z) acting on the hyperbolic plane reveals that the "piecewise hyperbolic" groups phi(Gp) here have prominent relatives: they are closely related to Richard Thompson's group V. And in the Euclidean case when G = Isom(Z^n) we find that the "piecewise Euclidean" groups pei(S) - as well as the corresponding "piecewise translation" groups pet(S) - have divers but to some extent accessible finiteness properties.

math.GR↗

arXiv2013

Infinite presentability of groups and condensation

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

math.GR↗

arXiv2012

On subsets of $S^n$ whose $(n+1)$-point subsets are contained in open hemisheres

We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri-Groves conjecture on cohomological finiteness conditions for metabelian groups. We find that there is a natural polyhedrality in a crucial special case. In the case of the two dimensional sphere we establish a strong polyhedrality condition for certain open sets which are maximal subject to satisfying the tameness condition that subsets of three or fewer points are contained in open hemispheres. Many examples are included.

math.GR↗

arXiv2009

Sigma Invariants of Direct Products of Groups

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}.

math.GR↗

arXiv2008

The Sigma Invariants of Thompson's Group F

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}.

math.GR↗

arXiv1998

Connectivity properties of group actions on non-positively curved spaces I: Controlled connectivity and openness results

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). Isometric actions of G on M are (by definition) points in the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems: 1. The cocompact actions form an open subset of R. 2. The cocompact actions with discrete orbits whose point-stabilizers have type F_n form an open subset of the subspace of R consisting of all actions with discrete orbits. (F_1 means finitely generated, F_2 means finitely presented etc.) The key idea is to introduce a new "controlled topology" invariant of such actions - dependent on n - which is unfamiliar when the orbits are not discrete but which becomes familiar (cf 2.) when the orbits are discrete. (This is the first of two papers.)

math.GR↗

arXiv1998

Connectivity properties of group actions on non-positively curved spaces II: The geometric invariants

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 bdM, both dependent on n. These subsets consist of those points of bdM at which - in two senses - the action is "controlled (n-1)-connected". One of these sets is open in the Tits metric topology. Even in classical cases like familiar groups of isometries of the hyperbolic plane or of a locally finite tree these sets seem to be new and interesting invariants. The "SIGMA-theory" of Bieri-Neumann-Strebel-Renz is recovered in the special case in which M is G(abelianized) tensor R with the translation action.

math.GR↗