86 research outputs found
Smooth group representations on bornological vector spaces
We develop the basic theory of smooth representations of locally compact
groups on bornological vector spaces. In this setup, we are able to formulate
better general theorems than in the topological case. Still, smooth
representations of totally disconnected groups on vector spaces and of Lie
groups on Frechet spaces remain special cases of our theory. We identify smooth
representations with essential modules over an appropriate convolution algebra.
We examine smoothening functors on representations and modules and show that
they agree if they are both defined. We establish the basic properties of
induction and compact induction functors using adjoint functor techniques. We
describe the center of the category of smooth representations.Comment: I corrected a mistake in the last section and added a french abstrac
Combable groups have group cohomology of polynomial growth
Group cohomology of polynomial growth is defined for any finitely generated
discrete group, using cochains that have polynomial growth with respect to the
word length function. We give a geometric condition that guarantees that it
agrees with the usual group cohomology and verify this condition for a class of
combable groups. Our condition involves a chain complex that is closely related
to exotic cohomology theories studied by Allcock and Gersten and by Mineyev.Comment: 19 pages, typo corrected in version
Homological algebra for Schwartz algebras of reductive p-adic groups
Let G be a reductive group over a non-Archimedean local field. Then the
canonical functor from the derived category of smooth tempered representations
of G to the derived category of all smooth representations of G is fully
faithful. Here we consider representations on bornological vector spaces. As a
consequence, if V and W are two tempered irreducible representations and if V
or W is square-integrable, then Ext_G^n(V,W) vanishes for all n>0. We use this
to prove in full generality a formula for the formal dimension of
square-integrable representations due to Schneider and Stuhler.Comment: 34 pages, version 2 contains, in addition, a discussion about formal
dimensions from the point of view of Schwartz algebras and von Neumann
algebra
Algebraic theory of vector-valued integration
We define a monad M on a category of measurable bornological sets, and we
show how this monad gives rise to a theory of vector-valued integration that is
related to the notion of Pettis integral. We show that an algebra X of this
monad is a bornological locally convex vector space endowed with operations
which associate vectors \int f dm in X to incoming maps f:T --> X and measures
m on T. We prove that a Banach space is an M-algebra as soon as it has a Pettis
integral for each incoming bounded weakly-measurable function. It follows that
all separable Banach spaces, and all reflexive Banach spaces, are M-algebras.Comment: shortened, e.g. by citing references regarding basic lemmas; made
changes to ordering of some lemmas and section
The isocohomological property, higher Dehn functions, and relatively hyperbolic groups
The property that the polynomial cohomology with coefficients of a finitely
generated discrete group is canonically isomorphic to the group cohomology is
called the (weak) isocohomological property for the group. In the case when a
group is of type , i.e. that has a classifying space with the
homotopy type of a cellular complex with finitely many cells in each dimension,
we show that the isocohomological property is equivalent to the universal cover
of the classifying space satisfying polynomially bounded higher Dehn functions.
If a group is hyperbolic relative to a collection of subgroups, each of which
is polynomially combable (respectively and isocohomological), then
we show that the group itself has these respective properties too. Combining
with the results of Connes-Moscovici and Dru{\c{t}}u-Sapir we conclude that a
group satisfies the Novikov conjecture if it is relatively hyperbolic to
subgroups that are of property RD, of type and isocohomological.Comment: 35 pages, no figure
Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories
Let G be a Lie group which is the union of an ascending sequence of Lie
groups G_n (all of which may be infinite-dimensional). We study the question
when G is the direct limit of the G_n's in the category of Lie groups,
topological groups, smooth manifolds, resp., topological spaces. Full answers
are obtained for G the group Diff_c(M) of compactly supported smooth
diffeomorphisms of a sigma-compact smooth manifold M, and for test function
groups C^infty_c(M,H) of compactly supported smooth maps with values in a
finite-dimensional Lie group H. We also discuss the cases where G is a direct
limit of unit groups of Banach algebras, a Lie group of germs of Lie
group-valued analytic maps, or a weak direct product of Lie groups.Comment: extended preprint version, 66 page
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