28 research outputs found
Isometric group actions on Banach spaces and representations vanishing at infinity
Our main result is that the simple Lie group acts properly
isometrically on if . To prove this, we introduce property
({\BP}_0^V), for be a Banach space: a locally compact group has
property ({\BP}_0^V) if every affine isometric action of on , such
that the linear part is a -representation of , either has a fixed point
or is metrically proper. We prove that solvable groups, connected Lie groups,
and linear algebraic groups over a local field of characteristic zero, have
property ({\BP}_0^V). As a consequence for unitary representations, we
characterize those groups in the latter classes for which the first cohomology
with respect to the left regular representation on is non-zero; and we
characterize uniform lattices in those groups for which the first -Betti
number is non-zero.Comment: 28 page
Property for noncommutative universal lattices
We establish a new spectral criterion for Kazhdan's property which is
applicable to a large class of discrete groups defined by generators and
relations. As the main application, we prove property for the groups
, where and is an arbitrary finitely generated
associative ring. We also strengthen some of the results on property for
Kac-Moody groups from a paper of Dymara and Januszkiewicz (Invent. Math 150
(2002)).Comment: 47 pages; final versio
Expansive actions on uniform spaces and surjunctive maps
We present a uniform version of a result of M. Gromov on the surjunctivity of
maps commuting with expansive group actions and discuss several applications.
We prove in particular that for any group and any field \K, the
space of -marked groups such that the group algebra \K[G] is
stably finite is compact.Comment: 21 page
Fixed points and amenability in non-positive curvature
Consider a proper cocompact CAT(0) space X. We give a complete algebraic
characterisation of amenable groups of isometries of X. For amenable discrete
subgroups, an even narrower description is derived, implying Q-linearity in the
torsion-free case.
We establish Levi decompositions for stabilisers of points at infinity of X,
generalising the case of linear algebraic groups to Is(X). A geometric
counterpart of this sheds light on the refined bordification of X (\`a la
Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is
further deduced that unimodular cocompact groups cannot fix any point at
infinity except in the Euclidean factor; this fact is needed for the study of
CAT(0) lattices.
Various fixed point results are derived as illustrations.Comment: 33 page
Continuity properties of measurable group cohomology
A version of group cohomology for locally compact groups and Polish modules
has previously been developed using a bar resolution restricted to measurable
cochains. That theory was shown to enjoy analogs of most of the standard
algebraic properties of group cohomology, but various analytic features of
those cohomology groups were only partially understood.
This paper re-examines some of those issues. At its heart is a simple
dimension-shifting argument which enables one to `regularize' measurable
cocycles, leading to some simplifications in the description of the cohomology
groups. A range of consequences are then derived from this argument.
First, we prove that for target modules that are Fr\'echet spaces, the
cohomology groups agree with those defined using continuous cocycles, and hence
they vanish in positive degrees when the acting group is compact. Using this,
we then show that for Fr\'echet, discrete or toral modules the cohomology
groups are continuous under forming inverse limits of compact base groups, and
also under forming direct limits of discrete target modules.
Lastly, these results together enable us to establish various circumstances
under which the measurable-cochains cohomology groups coincide with others
defined using sheaves on a semi-simplicial space associated to the underlying
group, or sheaves on a classifying space for that group. We also prove in some
cases that the natural quotient topologies on the measurable-cochains
cohomology groups are Hausdorff.Comment: 52 pages. [Nov 22, 2011:] Major re-write with Calvin C. Moore as new
co-author. Results from previous version strengthened and several new results
added. [Nov 25, 2012:] Final version now available at springerlink.co
Property (T) and rigidity for actions on Banach spaces
We study property (T) and the fixed point property for actions on and
other Banach spaces. We show that property (T) holds when is replaced by
(and even a subspace/quotient of ), and that in fact it is
independent of . We show that the fixed point property for
follows from property (T) when 1
. For simple Lie groups and their lattices, we prove that the fixed point property for holds for any if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement
C*-simple groups: amalgamated free products, HNN extensions, and fundamental groups of 3-manifolds
International audienc