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Functor homology over an additive category
We uncover several general phenomenas governing functor homology over
additive categories. In particular, we generalize the strong comparison theorem
of Franjou Friedlander Scorichenko and Suslin to the setting of Fp-linear
additive categories. Our results have a strong impact in terms of explicit
computations of functor homology, and they open the way to new applications to
stable homology of groups or to K-theory. As an illustration, we prove
comparison theorems between cohomologies of classical algebraic groups over
infinite perfect fields, in the spirit of a celebrated result of Cline,
Parshall, Scott et van der Kallen for finite fields
Spectral computations on lamplighter groups and Diestel-Leader graphs
The Diestel-Leader graph DL(q,r) is the horocyclic product of the homogeneous
trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of
the lamplighter group (wreath product of the cyclic group of order q with the
infinite cyclic group) with respect to a natural generating set. For the
"Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and
Dicks & Schick have determined the spectrum and the (on-diagonal) spectral
measure (Plancherel measure). Here, we show that thanks to the geometric
realization, these results can be obtained for all DL-graphs by directly
computing an l^2-complete orthonormal system of finitely supported
eigenfunctions of the SRW. This allows computation of all matrix elements of
the spectral resolution, including the Plancherel measure. As one application,
we determine the sharp asymptotic behaviour of the N-step return probabilities
of SRW. The spectral computations involve a natural approximating sequence of
finite subgraphs, and we study the question whether the cumulative spectral
distributions of the latter converge weakly to the Plancherel measure. To this
end, we provide a general result regarding Foelner approximations; in the
specific case of DL(q,r), the answer is positive only when r=q
K-theory for ring C*-algebras attached to function fields with only one infinite place
We study the K-theory of ring C*-algebras associated to rings of integers in
global function fields with only one single infinite place. First, we compute
the torsion-free part of the K-groups of these ring C*-algebras. Secondly, we
show that, under a certain primeness condition, the torsion part of K-theory
determines the inertia degrees at infinity of our function fields.Comment: 27 page
Cohomology of Artin groups of type tilde{A}_n, B_n and applications
We consider two natural embeddings between Artin groups: the group
G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type
B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} of
type A_n. The cohomologies of these groups are related, by standard results, in
a precise way. By using techniques developed in previous papers, we give
precise formulas (sketching the proofs) for the cohomology of G_{B_n} with
coefficients over the module Q[q^{+-1},t^{+-1}], where the action is
(-q)-multiplication for the standard generators associated to the first n-1
nodes of the Dynkin diagram, while is (-t)-multiplication for the generator
associated to the last node.
As a corollary we obtain the rational cohomology for G_{tilde{A}_n} as well
as the cohomology of Br_{n+1} with coefficients in the (n+1)-dimensional
representation obtained by Tong, Yang and Ma.
We stress the topological significance, recalling some constructions of
explicit finite CW-complexes for orbit spaces of Artin groups. In case of
groups of infinite type, we indicate the (few) variations to be done with
respect to the finite type case. For affine groups, some of these orbit spaces
are known to be K(pi,1) spaces (in particular, for type tilde{A}_n).
We point out that the above cohomology of G_{B_n} gives (as a module over the
monodromy operator) the rational cohomology of the fibre (analog to a Milnor
fibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.Comment: This is the version published by Geometry & Topology Monographs on 22
February 200
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