Proper Group Actions Research Articles

Proper actions of Grigorchuk groups on a CAT(0) cube complex

In this paper we will present a construction of a CAT(0) cube complex (an infinite cube), on which the uncountable family of Grigorchuk groups Gω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_\\omega $$\\end{document} act without bounded orbit. Moreover, if the sequence ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} does not contain repetition, we prove that the action is proper and faithful. As a consequence of this result, this cube complex is a model for the classifying space of proper actions for all the groups Gω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_\\omega $$\\end{document} with ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} without repetition. This construction works in a general way for any group acting on a set and which admits a commensurated subset. These examples of non-elliptic actions of infinite finitely generated torsion groups on a non-positively curved cube complex contrast to several established fixed-point theorems concerning actions of torsion groups.

The equivariant coarse Baum–Connes conjecture for metric spaces with proper group actions

The equivariant coarse Baum–Connes conjecture interpolates between the Baum–Connes conjecture for a discrete group and the coarse Baum–Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group \Gamma acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/\Gamma admits a coarse embedding into Hilbert space and \Gamma is amenable, and that the \Gamma -orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum–Connes conjecture holds for (X,\Gamma) . Along the way, we prove a K -theoretic amenability statement for the \Gamma -space X under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K -theory.

Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about the relationship between mapping class groups and braid groups, we discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface $M$ with a free and proper group action in this article. Our construction is based on the exact sequence given by the fibration $\mathcal{F}_{0}^{G}M \rightarrow F(M/G,n)$. The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is $\mathbb{T}^{2}$.

Equivariant Callias index theory via coarse geometry

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the $K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the $K$-theory of a group $C^*$-algebra.

Criterion of proper actions of abelian affine discontinuous groups for ℝn

In [Formula: see text], Auslander conjectured that every crystallographic subgroup [Formula: see text] of the affine group [Formula: see text] acting properly discontinuously on [Formula: see text] is virtually solvable, i.e. contains a solvable subgroup of finite index. One of the major difficulties to prove such a conjecture is to generate a criterion of the proper action of a given discrete group acting properly discontinuously on [Formula: see text]. In this paper, we focus on this question in the setting where [Formula: see text] is abelian. We show that [Formula: see text] acts properly on [Formula: see text] if and only if any [Formula: see text]-orbit is discrete (or equivalelntly closed). We further generate an equivalent criterion in the setting where [Formula: see text] stands for a crystallographic discontinuous affine group for [Formula: see text] based upon a generating family.

The completion theorem in twisted equivariant K-theory for proper actions

We compare different algebraic structures in twisted equivariant K-theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-theory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant K-theory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel K-homology for discrete groups.

On invariant linearization of Lie groupoids

The linearization theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization remains somehow open. We address it here, by first giving a counter-example to a previous conjecture, and then proving a sufficient criterion that uses compatible complete metrics and covers the case of proper group actions. We also show a partial converse that fixes and extends previous results in the literature.

Generic free subgroups and statistical hyperbolicity

This paper studies the generic behavior of k -tuples of elements for k \geq 2 in a proper group action with contracting elements, with applications toward relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convex-cocompact action, we show that an exponential generic set of k elements for any fixed k \geq 2 generates a quasi-isometrically embedded free subgroup of rank k . For k = 2 , we study the sprawl property of group actions and establish that statistically convex-cocompact actions are statistically hyperbolic in the sense of M. Duchin, S. Lelièvre, and C. Mooney. For any proper action with a contracting element, if it satisfies a condition introduced by Dal’bo-Otal-Peigné and has purely exponential growth, we obtain the same results on generic free subgroups and statistical hyperbolicity.

Equivariant higher-index problems for proper actions and nonpositively curved manifolds

We prove that the equivariant higher-index map is injective for a bounded geometry metric space with a proper and isometric group action which can be equivariantly embedded into a simply connected, complete, and nonpositively curved Riemannian manifold with a proper and isometric group action.

Twisted geometric K-homology for proper actions of discrete groups

We define Twisted Equivariant [Formula: see text]-homology groups using geometric cycles. We compare them with analytical approaches using Kasparov KK-theory and (twisted) [Formula: see text]-algebras of groups and groupoids.

Orbispaces as differentiable stratified spaces

We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures—it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek’s framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we extend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complementary exposition to those available and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions.

Suborbifolds, quotients and transversality

Inspired by work of Borzellino and Brunsden, we generalize the notion of a submanifold identifying a natural and sufficiently general condition which guarantees that a subset of an (effective) orbifold carries itself a canonical induced orbifold structure. We illustrate the strength of this approach generalizing typical constructions of submanifolds to the orbifold setting using embeddings, proper group actions and the idea of transversality.

Elliptic and transversally elliptic index theory from the viewpoint of $KK$-theory

The main purpose of the paper is to develop a KK -theory approach to index theory of operators transversally elliptic with respect to a proper group action. The principal novelty is the use of fields of Clifford algebras along the orbits. Index formulas that we prove allow to calculate index from symbol and vice-versa using KK -product. This has a deeper meaning: there exists Poincaré duality between the symbol and the index when they are defined as elements of the appropriate KK -groups. The main results on transversally elliptic operators are contained in the second half of the paper. In the first part we develop the necessary technique and prove index theorems for elliptic operators. This is done in the equivariant setting with respect to a proper group action, the manifolds and groups are not assumed to be compact. At the end of the paper we give applications of our methods to transversally elliptic operators on singular foliations.

Fixed point property for a CAT(0) space which admits a proper cocompact group action

We prove that if a geodesically complete $\mathrm{CAT}(0)$ space $X$ admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of $X$ is less than $1$. Let $G$ be a finite connected graph, $\mu_1 (G)$ be the linear spectral gap of $G$, and $\lambda_1 (G,X)$ be the nonlinear spectral gap of $G$ with respect to such a $\mathrm{CAT}(0)$ space $X$. Then, the result implies that the ratio $\lambda_1 (G,X) / \mu_1 (G)$ is bounded from below by a positive constant which is independent of the graph $G$. It follows that any isometric action of a random group of the graph model on such $X$ has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.

Spin-structures and proper group actions

We generalise Atiyah and Hirzebruch's vanishing theorem for actions by compact groups on compact Spin-manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact Spin-manifolds. As corollaries, we obtain some vanishing results for an Aˆ-type genus.

The index theory on non-compact manifolds with proper group action

We construct a regularized index of a generalized Dirac operator on a complete Riemannian manifold endowed with a proper action of a unimodular Lie group. We show that the index is preserved by a certain class of non-compact cobordisms and prove a gluing formula for the regularized index. The results of this paper generalize our previous construction of index for compact group action and the recent paper of Hochs and Mathai who studied the case of a Hamiltonian action on a symplectic manifold. As an application of the cobordism invariance of the index we give an affirmative answer to a question of Hochs and Mathai about the independence of the Hochs–Mathai quantization of the metric, connection and other choices.

Proper actions of high-dimensional groups on complex manifolds

Proper group actions are ubiquitous in mathematics and have many of the attractive features of actions of compact groups. In this survey, we discuss proper actions of Lie groups on smooth manifolds. If the group dimension is sufficiently high, all proper effective actions can be explicitly determined, and our principal goal is to provide a comprehensive exposition of known classification results in the complex setting. They include a complete description of Kobayashi-hyperbolic manifolds with high-dimensional automorphism group, which is a case of special interest.

A restriction on proper group actions on homogeneous spaces of reductive type

Let $$L$$ be a reductive subgroup of a reductive Lie group $$G$$ . Let $$G/H$$ be a homogeneous space of reductive type. We provide a necessary condition for the properness of the action of $$L$$ on $$G/H$$ . As an application we give examples of spaces that do not admit standard compact Clifford-Klein forms.

Weakly proper group actions, Mansfield’s Imprimitivity and twisted Landstad duality

Using the theory of weakly proper actions of locally compact groups recently developed by the authors, we give a unified proof of both reduced and maximal versions of Mansfield’s Imprimitivity Theorem and obtain a general version of Landstad’s Duality Theorem for twisted group coactions. As one application, we obtain the stabilization trick for arbitrary twisted coactions, showing that every twisted coaction is Morita equivalent to an inflated coaction.

Dimension invariants for groups admitting a cocompact model for proper actions

Abstract Let G be a group that admits a cocompact classifying space for proper actions X. We derive a formula for the Bredon cohomological dimension for proper actions of G in terms of the relative cohomology with compact support of certain pairs of subcomplexes of X. We use this formula to compute the Bredon cohomological dimension for proper actions of fundamental groups of non-positively curved simple complexes of finite groups. As an application we show that if a virtually torsion-free group acts properly and chamber transitively on a building, its virtual cohomological dimension coincides with its Bredon cohomological dimension. This covers the case of Coxeter groups and graph products of finite groups.