205 research outputs found
Geodesic flow for CAT(0)-groups
We associate to a CAT(0)-space a flow space that can be used as the
replacement for the geodesic flow on the sphere tangent bundle of a Riemannian
manifold. We use this flow space to prove that CAT(0)-group are transfer
reducible over the family of virtually cyclic groups. This result is an
important ingredient in our proof of the Farrell-Jones Conjecture for these
groups
The L^2-Alexander torsion of 3-manifolds
We introduce -Alexander torsions for 3-manifolds, which can be viewed as
a generalization of the -Alexander polynomial of Li--Zhang. We state the
-Alexander torsions for graph manifolds and we partially compute them for
fibered manifolds. We furthermore show that given any irreducible 3-manifold
there exists a coefficient system such that the corresponding -torsion
detects the Thurston norm.Comment: 47 pages v3: fixed many typos, updated references and improved the
exposition, following the referees suggestion
Euler Characteristics of Categories and Homotopy Colimits
In a previous article, we introduced notions of finiteness obstruction, Euler
characteristic, and L^2-Euler characteristic for wide classes of categories. In
this sequel, we prove the compatibility of those notions with homotopy colimits
of I-indexed categories where I is any small category admitting a finite
I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit
Formula include formulas for products, homotopy pushouts, homotopy orbits, and
transport groupoids. We also apply our formulas to Haefliger complexes of
groups, which extend Bass--Serre graphs of groups to higher dimensions. In
particular, we obtain necessary conditions for developability of a finite
complex of groups from an action of a finite group on a finite category without
loops.Comment: 44 pages. This final version will appear in Documenta Mathematica.
Remark 8.23 has been improved, discussion of Grothendieck construction has
been slightly expanded at the beginning of Section 3, and a few other minor
improvements have been incoporate
Finiteness obstructions and Euler characteristics of categories
We introduce notions of finiteness obstruction, Euler characteristic,
L^2-Euler characteristic, and M\"obius inversion for wide classes of
categories. The finiteness obstruction of a category Gamma of type (FP) is a
class in the projective class group K_0(RGamma); the functorial Euler
characteristic and functorial L^2-Euler characteristic are respectively its
RGamma-rank and L^2-rank. We also extend the second author's K-theoretic
M\"obius inversion from finite categories to quasi-finite categories. Our main
example is the proper orbit category, for which these invariants are
established notions in the geometry and topology of classifying spaces for
proper group actions. Baez-Dolan's groupoid cardinality and Leinster's Euler
characteristic are special cases of the L^2-Euler characteristic. Some of
Leinster's results on M\"obius-Rota inversion are special cases of the
K-theoretic M\"obius inversion.Comment: Final version, accepted for publication in the Advances in
Mathematics. Notational change: what was called chi(Gamma) in version 1 is
now called chi(BGamma), and chi(Gamma) now signifies the sum of the
components of the functorial Euler characteristic chi_f(Gamma). Theorem 5.25
summarizes when all Euler characteristics are equal. Minor typos have been
corrected. 88 page
The Ore condition, affiliated operators, and the lamplighter group
Let G be the wreath product of Z and Z/2, the so called lamplighter group and
k a commutative ring. We show that kG does not have a classical ring of
quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka
notebook problem. Assume that kG is contained in a ring R in which the element
1-x is invertible, with x a generator of Z considered as subset of G. Then R is
not flat over kG. If k is the field of complex numbers, this applies in
particular to the algebra UG of unbounded operators affiliated to the group von
Neumann algebra of G. We present two proofs of these results. The second one is
due to Warren Dicks, who, having seen our argument, found a much simpler and
more elementary proof, which at the same time yielded a more general result
than we had originally proved. Nevertheless, we present both proofs here, in
the hope that the original arguments might be of use in some other context not
yet known to us.Comment: LaTex2e, 7 pages. Added a new proof of the main result (due to Warren
Dicks) which is shorter, easier and more elementary, and at the same time
yields a slightly more general result. Additionally: misprints removed. to
appear in Proceedings of "Higher dimensional manifold theory", Conference at
ICTP Trieste 200
Equivariant Euler characteristics and K-homology Euler classes for proper cocompact G-manifolds
Let G be a countable discrete group and let M be a smooth proper cocompact
G-manifold without boundary. The Euler operator defines via Kasparov theory an
element, called the equivariant Euler class, in the equivariant K-homology of
M. The universal equivariant Euler characteristic of M, which lives in a group
U^G(M), counts the equivariant cells of M, taking the component structure of
the various fixed point sets into account. We construct a natural homomorphism
from U^G(M) to the equivariant KO-homology of M. The main result of this paper
says that this map sends the universal equivariant Euler characteristic to the
equivariant Euler class. In particular this shows that there are no `higher'
equivariant Euler characteristics. We show that, rationally, the equivariant
Euler class carries the same information as the collection of the orbifold
Euler characteristics of the components of the L-fixed point sets M^L, where L
runs through the finite cyclic subgroups of G. However, we give an example of
an action of the symmetric group S_3 on the 3-sphere for which the equivariant
Euler class has order 2, so there is also some torsion information.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper16.abs.htm
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