Torsion-free Discrete Group Research Articles

On Nielsen realization and manifold models for classifying spaces

We consider the problem of whether, for a given virtually torsionfree discrete group Γ \Gamma , there exists a cocompact proper topological Γ \Gamma -manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that Γ \Gamma contains a normal torsionfree subgroup π \pi such that π \pi is hyperbolic and π \pi is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and Γ / π \Gamma /\pi is a finite cyclic group of odd order.

Δ-weakly mixing subset in positive entropy actions of a nilpotent group

The notion of Δ-weakly mixing subsets is introduced for countable torsion-free discrete group actions. It is shown that for a finitely generated torsion-free discrete nilpotent group action, positive topological entropy implies the existence of Δ-weakly mixing subsets, and while there exists a finitely generated torsion-free discrete solvable group action which has positive topological entropy but without any Δ-weakly mixing subsets.

Index theory for manifolds with Baas–Sullivan singularities

We study index theory for manifolds with Baas–Sullivan singularities using geometric K -homology with coefficients in a unital C^* -algebra. In particular, we define a natural analog of the Baum–Connes assembly map for a torsion-free discrete group in the context of these singular spaces. The cases of singularities modelled on k -points (i.e., $\mathbb Z/k\mathbb Z-manifolds) and the circle are discussed in detail. In the case of the former, the associated index theorem is related to the Freed–Melrose index theorem; in the case of the latter, the index theorem is related to work of Rosenberg.

Centralizers in à 2 groups

Abstract Let Γ be a torsion-free discrete group acting cocompactly on a two dimensional euclidean building Δ. The centralizer of an element of Γ is either a Bieberbach group or is described by a finite graph of finite cyclic groups. Explicit examples are computed, with Δ of type à 2.

An equivariant higher index theory and nonpositively curved manifolds

In this paper, we define an equivariant higher index map from K ∗ Γ ( X ) to K ∗ ( C ∗ ( X ) Γ ) if a torsion-free discrete group Γ acts on a manifold X properly, where C ∗ ( X ) Γ is the norm closure of all locally compact, Γ-invariant operators with finite propagation. When Γ acts on X properly and cocompactly, this equivariant higher index map coincides with the Baum–Connes map [P. Baum, A. Connes, K-theory for discrete groups, in: D. Evens, M. Takesaki (Eds.), Operator Algebras and Applications, Cambridge Univ. Press, Cambridge, 1989, pp. 1–20; P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C ∗ -algebras, in: C ∗ -Algebras: 1943–1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291]. When Γ is trivial, this equivariant higher index map is the coarse Baum–Connes map [J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (497) (1993); J. Roe, Index Theory, Coarse Geometry, and the Topology of Manifolds, CBMS Reg. Conf. Ser. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1996]. If X is a simply-connected complete Riemannian manifold with nonpositive sectional curvature and Γ is a torsion-free discrete group acting on X properly and isometrically, we prove that the equivariant higher index map is injective.

Causal properties of AdS-isometry groups I: causal actions and limit sets

We study the causality relation in the $3$-dimensional anti-de Sitter space AdS and its conformal boundary $\mbox{Ein}_2$. To any closed achronal subset $\Lambda$ in $\mbox{Ein}_2$ we associate the invisible domain $E(\Lambda)$ from $\Lambda$ in AdS. We show that if $\Gamma$ is a torsion-free discrete group of isometries of AdS preserving $\Lambda$ and is non-elementary (for example, not abelian) then the action of $\Gamma$ on $E(\Lambda)$ is free, properly discontinuous and strongly causal. If $\Lambda$ is a topological circle then the quotient space $M_\Lambda(\Gamma) = \Gamma\backslash{E}(\Lambda)$ is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface $S$ such that the induced metric on $S$ is complete. In a forthcoming paper we study the case where $\Gamma$ is elementary and use the results of the present paper to define a large family of AdS-spacetimes including all the previously known examples of BTZ multi-black holes.

A descent principle for the Dirac–dual-Dirac method

Let G be a torsion-free discrete group with a finite-dimensional classifying space B G . We show that G has a dual-Dirac morphism if and only if a certain coarse (co-)assembly map is an isomorphism. Hence the existence of a dual-Dirac morphism for such groups is a metric, that is, coarse, invariant. We get results for groups with torsion as well.

EZ-structures and topological applications

In this paper, we introduce the notion of an EZ-structure on a group, an equivariant version of the Z-structures introduced by Bestvina [4]. Examples of groups having an EZ-structure include (1) torsion free \delta -hyperbolic groups, and (2) torsion free \mathrm{CAT}(0) -groups. Our first theorem shows that any group having an EZ-structure has an action by homeomorphisms on some (\mathbb D^n, \Delta) , where n is sufficiently large, and \Delta is a closed subset of \partial \mathbb D^n=S^{n-1} . The action has the property that it is proper and cocompact on \mathbb D^n-\Delta , and that if K\subset \mathbb D^n-\Delta is compact, that \mathrm{diam}(gK) tends to zero as g\rightarrow \infty . We call this property (*_\Delta) . Our second theorem uses techniques of Farrell–Hsiang [8] to show that the Novikov conjecture holds for any torsion-free discrete group satisfying condition (*_\Delta) (giving a new proof that torsion-free \delta -hyperbolic and \mathrm{CAT}(0) groups satisfy the Novikov conjecture). Our third theorem gives another application of our main result. We show how, in the case of a torsion-free \delta -hyperbolic group \Gamma , we can obtain a lower bound for the homotopy groups \pi_n(\mathcal P(B\Gamma)) , where \mathcal P(\cdot ) is the stable topological pseudo-isotopy functor.

On the Validity of Failure of Gap Rigidity for Certain Pairs of Bounded Symmetric Domains

Let Ω be a bounded symmetric domain equipped with a canonical Kahler metric. We are interested to characterize holomorphic geodesic cycles (i.e., compact complex geodesic submanifolds) S ⊂ X on Hermitian locally symmetric manifolds X uniformized by Ω, i.e., X = Ω/Γ, where Γ ⊂ Aut(Ω) is any torsion-free discrete group of automorphisms, in terms of differentialgeometric or algebro-geometric conditions. In Eyssidieux-Mok [EysMok1995] we studied almost geodesic complex submanifolds and formulated the gap phenomenon. Up to equivalence under Aut(Ω) there are only a finite number of totally geodesic complex submanifolds D ⊂ Ω which are themselves biholomorpic to bounded symmetric domains. Let 2 > 0. We say that S ⊂ X is 2-geodesic if and only if the norm of the second fundamental form of S ⊂ X is uniformly bounded by 2. Fixing Ω but letting Γ ⊂ Ω be arbitrary we showed that when 2 is sufficiently small, an 2-geodesic compact complex

The Kadison-Kaplansky conjecture for word-hyperbolic groups

In this paper we prove the Kadison-Kaplansky idempotent conjecture for torsion-free word-hyperbolic groups. The conjecture asserts that the following equivalent statements hold for a torsion-free discrete group Γ: • The reduced group C∗-algebra C∗ r (Γ) contains no idempotents except 0 and 1. • The spectrum of every element of the reduced group C∗-algebra is connected. • The canonical trace on C∗ r (Γ) takes integer values on idempotents. The last assertion can be viewed as a statement about the pairing between the K-theory and the (local) cyclic cohomology of the group C∗-algebra. It is in this setting that we will prove the conjecture. Our proof is based on a partial analysis of the assembly maps in K-theory and local cyclic homology. We compare these assembly maps by means of an equivariant bivariant Chern-Connes character. Before going into details, we recall some previous work on the conjecture. The first progress was achieved by Pimsner and Voiculescu [PV] who proved the Kadison-Kaplansky conjecture for free groups as a consequence of their computation of the K-theory of the group C∗-algebra. Subsequently it was realized that more generally the Kadison-Kaplansky conjecture was a consequence of the Baum-Connes conjecture which gives a geometric description of K∗(C∗ r (Γ)) for any torsion-free discrete group. In fact the Baum-Connes conjecture states that the K-theoretic assembly map

Unitary Measures on LCA Groups

A unitary measure on a locally compact Abelian (LCA) group G is a complex measure whose Fourier transform is of absolute value 1 everywhere. The problem of finding all such measures is known to be closely related to that of finding all invertible measures on G. In this paper, we find all unitary measures when G is the circle or a discrete group. If G is a torsion-free discrete group, the characterization generalizes a theorem of Bohr.

Unitary measures on LCA groups

A unitary measure on a locally compact Abelian (LCA) group G is a complex measure whose Fourier transform is of absolute value 1 everywhere. The problem of finding all such measures is known to be closely related to that of finding all invertible measures on G. In this paper, we find all unitary measures when G is the circle or a discrete group. If G is a torsion-free discrete group, the characterization generalizes a theorem of Bohr.

Real Borel cohomology of locally compact groups

enemy. Some of the theorems in the first two sections are modeled on those of Snapper, in [18], who considered these problems for everything discrete, the powerful techniques of homological algebra being available in this case. Unfortunately, there is no machinery of projective or injective resolutions immediately at our disposal for the cohomology theories based on Borel cochains. There is a type of resolution, which leans toward the injective, but its full homological character is not understood as yet. The entire idea of considering the cohomology of general group actions was inspired by the paper of Snapper. These things have been considered in the continuous case by Mostow in [17], where equivariant cohomology groups are defined and studied. Almost nothing being known about the higher cohomology groups, it was desirable to ferret out those isolated pockets of triviality. This is done in the last section, where it is shown that for G abelian locally compact separable metrizable, R the real line, G operating trivially on R, Hn(G, R), the bounded Borel groups defined in [16], do not depend upon the topology of G, nor its torsion. In fact, R seems to ignore everything about G except a certain torsion-free discrete group D associated to G. The further fact that these groups are isomorphic to the group of alternating multilinear functions An (G, R) was suggested by the results of Kleppner on H2(G, T), where G is locally compact abelian and T the circle group. He showed that for all G except a certain intractable class of 2-primary groups, whose fate is yet undecided, that H2(G, T)=L2(G, T)/S2(G, T), the continuous bihomomorphisms modulo the symmetric ones. We have not resolved this open question,