22 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
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
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
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
Quotation via Dialogical Interaction
International audienceQuotation has been much studied in philosophy. Given that quotation allows one to diagonalize out of any grammar, there have been comparatively few attempts within the linguistic literature to develop an account within a formal linguistic theory. Nonetheless, given the ubiquity of quotation in natural language, linguists need to explicate the formal mechanisms it employs. The central claim of this paper is that once one assumes a dialogical perspective on language such as provided by the KoS 1 framework, formalized in a rich type theory like Type Theory with Records (TTR), much of the mystery evaporates. In particular, one can utilize as denotations for quotative constructions entities that are independently motivated for dialogue processing—utterance types and locutionary propositions, austinian propositions about speech events