183 research outputs found
Groups of piecewise projective homeomorphisms
The group of piecewise projective homeomorphisms of the line provides
straightforward counter-examples to the so-called von Neumann conjecture. The
examples are so simple that many additional properties can be established.Comment: This version submitted to PNAS on October 22, 2012. Final version
published in PNAS at http://dx.doi.org/10.1073/pnas.121842611
Equivariant measurable liftings
We discuss equivariance for linear liftings of measurable functions.
Existence is established when a transformation group acts amenably, as e.g. the
Moebius group of the projective line.
Since the general proof is very simple but not explicit, we also provide a
much more explicit lifting for semi-simple Lie groups acting on their
Furstenberg boundary, using unrestricted Fatou convergence. This setting is
relevant to cocycles for characteristic classes.Comment: Removed the sigma-compactness assumption from Theorem A; minor
correction
Product groups acting on manifolds
We analyse volume-preserving actions of product groups on Riemannian
manifolds. To this end, we establish a new superrigidity theorem for ergodic
cocycles of product groups ranging in linear groups. There are no a priori
assumptions on the acting groups, except a spectral gap assumption on their
action.
Our main application to manifolds concerns irreducible actions of Kazhdan
product groups. We prove the following dichotomy: Either the action is
infinitesimally linear, which means that the derivative cocycle arises from
unbounded linear representations of all factors. Otherwise, the action is
measurably isometric, in which case there are at most two factors in the
product group.
As a first application, this provides lower bounds on the dimension of the
manifold in terms of the number of factors in the acting group. Another
application is a strong restriction for actions of non-linear groups.Comment: To appear in the Duke Mathematical Journal; 32 pages. Minor
revisions, including the addition of a variation on Theorem
The cup product of Brooks quasimorphisms
We prove the vanishing of the cup product of the bounded cohomology classes
associated to any two Brooks quasimorphisms on the free group. This is a
consequence of the vanishing of the square of a universal class for tree
automorphism groups.Comment: 7 page
An exotic deformation of the hyperbolic space
On the one hand, we construct a continuous family of non-isometric proper
CAT(-1) spaces on which the isometry group of the
real hyperbolic -space acts minimally and cocompactly. This provides the
first examples of non-standard CAT(0) model spaces for simple Lie groups.
On the other hand, we classify all continuous non-elementary actions of on the infinite-dimensional real hyperbolic space. It
turns out that they are in correspondence with the exotic model spaces that we
construct.Comment: 42 pages, minor modifications, this is the final versio
The Dixmier problem, lamplighters and Burnside groups
J. Dixmier asked in 1950 whether every non-amenable group admits uniformly
bounded representations that cannot be unitarised. We provide such
representations upon passing to extensions by abelian groups. This gives a new
characterisation of amenability. Furthermore, we deduce that certain Burnside
groups are non-unitarisable, answering a question raised by G. Pisier.Comment: 4 page
On the topological full group of a minimal Cantor Z^2-system
Grigorchuk and Medynets recently announced that the topological full group of
a minimal Cantor Z-action is amenable. They asked whether the statement holds
for all minimal Cantor actions of general amenable groups as well. We answer in
the negative by producing a minimal Cantor Z^2-action for which the topological
full group contains a non-abelian free group.Comment: 3 pages (some comments added
The norm of the Euler class
We prove that the norm of the Euler class E for flat vector bundles is
(in even dimension , since it vanishes in odd dimension). This
shows that the Sullivan--Smillie bound considered by Gromov and Ivanov--Turaev
is sharp. We construct a new cocycle representing E and taking only the two
values ; a null-set obstruction prevents any cocycle from existing
on the projective space. We establish the uniqueness of an antisymmetric
representative for E in bounded cohomology.Comment: 19 page
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