2,178 research outputs found
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
Coarse topology, enlargeability, and essentialness
Using methods from coarse topology we show that fundamental classes of closed
enlargeable manifolds map non-trivially both to the rational homology of their
fundamental groups and to the K-theory of the corresponding reduced
C*-algebras. Our proofs do not depend on the Baum--Connes conjecture and
provide independent confirmation for specific predictions derived from this
conjecture.Comment: 21 pages, 2 figures. Revised version. To appear in Ann. Sci. Ecole
Norm. Su
The strong Atiyah conjecture for right-angled Artin and Coxeter groups
We prove the strong Atiyah conjecture for right-angled Artin groups and
right-angled Coxeter groups. More generally, we prove it for groups which are
certain finite extensions or elementary amenable extensions of such groups.Comment: Minor change
On a conjecture of Atiyah
In this note we explain how the computation of the spectrum of the
lamplighter group from \cite{Grigorchuk-Zuk(2000)} yields a counterexample to a
strong version of the Atiyah conjectures about the range of -Betti numbers
of closed manifolds.Comment: 8 pages, A4 pape
The strong Novikov conjecture for low degree cohomology
We show that for each discrete group G, the rational assembly map
K_*(BG) \otimes Q \to K_*(C*_{max} G) \otimes \Q is injective on classes dual
to the subring generated by cohomology classes of degree at most 2 (identifying
rational K-homology and homology via the Chern character). Our result implies
homotopy invariance of higher signatures associated to these cohomology
classes. This consequence was first established by Connes-Gromov-Moscovici and
Mathai.
Our approach is based on the construction of flat twisting bundles out of
sequences of almost flat bundles as first described in our previous work. In
contrast to the argument of Mathai, our approach is independent of (and indeed
gives a new proof of) the result of Hilsum-Skandalis on the homotopy invariance
of the index of the signature operator twisted with bundles of small curvature.Comment: 11 page
A K-Theoretic Proof of Boutet de Monvel's Index Theorem for Boundary Value Problems
We study the C*-closure A of the algebra of all operators of order and class
zero in Boutet de Monvel's calculus on a compact connected manifold X with
non-empty boundary. We find short exact sequences in K-theory
0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K
denotes the compact ideal and T*X' the cotangent bundle of the interior of X.
Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we
show that the Fredholm index of an elliptic element in A is given as the
composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X'))
defined above. This relation was first established by Boutet de Monvel by
different methods.Comment: Title slightly changed. Accepted for publication in Journal fuer die
reine und angewandte Mathemati
Kulturschutz – Arbeitszeitbedarf beim Einsatz von Netzen
The product quality requirements of modern outdoor vegetable cultivation are becoming
more stringent and increasing importance is also attached to a defined guarantee
of yield. These demands may be satisfied by using protective netting.
The working methods employed in practice and the process technology hardly differ.
The differences between individual variants - 1 to 7% per net – are therefore relatively
low, as is the size degression. An average of 1.3 manpower-minutes per m are required
for the complete process. The time requirement per net is about 26.7 min., 13 min.
of which are needed for placement and 11.1 min. for removal. The difference is made
up by transportation to and from the field and loading the nets.
Covering the crop takes up 48 % or almost half the total working time requirement.
Removing the nets accounts for 42 %. 10 % is used in driving to and from the field for
placement and in loading the rolls or bales of netting before placement and after
removal. There is no additional transportation time, as the nets are carried as part of
the relevant transport to and from the field. Removal involves no extra journey times if
the netting is removed from the crop immediately prior to harvesting, the journey to
and from the field then being calculated as part of the working time requirement for
harvesting
Coarse homotopy groups
In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex.
To prove this we develop geometric triangulation techniques for cones which we expect to be of relevance also in different contexts
- …