4,909 research outputs found
Finitely generated nilpotent group C*-algebras have finite nuclear dimension
We show that group C*-algebras of finitely generated, nilpotent groups have
finite nuclear dimension. It then follows, from a string of deep results, that
the C*-algebra generated by an irreducible representation of such a group
has decomposition rank at most 3. If, in addition, satisfies the universal
coefficient theorem, another string of deep results shows it is classifiable by
its Elliott invariant and is approximately subhomogeneous. We give a large
class of irreducible representations of nilpotent groups (of arbitrarily large
nilpotency class) that satisfy the universal coefficient theorem and therefore
are classifiable and approximately subhomogeneous.Comment: Fixed typos. Question 5.1 of the previous version was already
answered in the literature; we have provided the appropriate referenc
Nilpotence and descent in equivariant stable homotopy theory
Let be a finite group and let be a family of subgroups of
. We introduce a class of -equivariant spectra that we call
-nilpotent. This definition fits into the general theory of
torsion, complete, and nilpotent objects in a symmetric monoidal stable
-category, with which we begin. We then develop some of the basic
properties of -nilpotent -spectra, which are explored further
in the sequel to this paper.
In the rest of the paper, we prove several general structure theorems for
-categories of module spectra over objects such as equivariant real and
complex -theory and Borel-equivariant . Using these structure theorems
and a technique with the flag variety dating back to Quillen, we then show that
large classes of equivariant cohomology theories for which a type of
complex-orientability holds are nilpotent for the family of abelian subgroups.
In particular, we prove that equivariant real and complex -theory, as well
as the Borel-equivariant versions of complex-oriented theories, have this
property.Comment: 63 pages. Revised version, to appear in Advances in Mathematic
Residual properties of 3-manifold groups I: Fibered and hyperbolic 3-manifolds
Let be a prime. In this paper, we classify the geometric 3-manifolds
whose fundamental groups are virtually residually . Let be a
virtually fibered 3-manifold. It is well-known that is residually
solvable and even residually finite solvable. We prove that is always
virtually residually . Using recent work of Wise, we prove that every
hyperbolic 3-manifold is either closed or virtually fibered and hence has a
virtually residually fundamental group. We give some generalizations to
pro- completions of groups, mapping class groups, residually torsion-free
nilpotent 3-manifold groups and central extensions of residually groups.Comment: 25 pages. Complete rewrit
Finite group extensions and the Atiyah conjecture
The Atiyah conjecture for a discrete group G states that the -Betti
numbers of a finite CW-complex with fundamental group G are integers if G is
torsion-free and are rational with denominators determined by the finite
subgroups of G in general. Here we establish conditions under which the Atiyah
conjecture for a group G implies the Atiyah conjecture for every finite
extension of G. The most important requirement is that the cohomology
is isomorphic to the cohomology of the p-adic completion
of G for every prime p. An additional assumption is necessary, e.g. that the
quotients of the lower central series or of the derived series are
torsion-free. We prove that these conditions are fulfilled for a class of
groups which contains Artin's pure braid groups, free groups, surfaces groups,
certain link groups and one-relator groups. Therefore every finite, in fact
every elementary amenable extension of these groups satisfies the Atiyah
conjecture. In the course of the proof we prove that if these extensions are
torsion-free, then they have plenty of non-trivial torsion-free quotients which
are virtually nilpotent. All of this applies in particular to Artin's full
braid group, therefore answering question B6 on http://www.grouptheory.info .
Our methods also apply to the Baum-Connes conjecture. This is discussed in
arXiv:math/0209165 "Finite group extensions and the Baum-Connes conjecture",
where the Baum-Connes conjecture is proved e.g. for the full braid group.Comment: 54 pages, typos and small mistakes corrected, final version to appear
in Journal of the AM
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