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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
Left-orderings on free products of groups
We show that no left-ordering on a free product of (left-orderable) groups is
isolated. In particular, we show that the space of left-orderings of free
product of finitely generated groups is homeomorphic to the Cantor set. With
the same techniques, we also give a new and constructive proof of the fact that
the natural conjugation action of the free group (on two or more generators) on
its space of left-orderings has a dense orbit.Comment: 13 page
Centralizers of maximal regular subgroups in simple Lie groups and relative congruence classes of representations
In the paper we present a new, uniform and comprehensive description of
centralizers of the maximal regular subgroups in compact simple Lie groups of
all types and ranks. The centralizer is either a direct product of finite
cyclic groups, a continuous group of rank 1, or a product, not necessarily
direct, of a continuous group of rank 1 with a finite cyclic group. Explicit
formulas for the action of such centralizers on irreducible representations of
the simple Lie algebras are given.Comment: 27 page
Ewens measures on compact groups and hypergeometric kernels
On unitary compact groups the decomposition of a generic element into product
of reflections induces a decomposition of the characteristic polynomial into a
product of factors. When the group is equipped with the Haar probability
measure, these factors become independent random variables with explicit
distributions. Beyond the known results on the orthogonal and unitary groups
(O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family
of probability changes analogous to the biassing in the Ewens sampling formula
known for the symmetric group. Then we study the spectral properties of these
measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The
associated orthogonal polynomials give rise, as tends to infinity to a
limit kernel at the singularity.Comment: New version of the previous paper "Hua-Pickrell measures on general
compact groups". The article has been completely re-written (the presentation
has changed and some proofs have been simplified). New references added
Nielsen Realisation by Gluing: Limit Groups and Free Products
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation
theorems from the setting of free groups to that of free products. As a result,
we obtain a fixed point theorem for finite groups of outer automorphisms acting
on the relative free splitting complex of Handel--Mosher and on the outer space
of a free product of Guirardel--Levitt, as well as a relative version of the
Nielsen realisation theorem, which in the case of free groups answers a
question of Karen Vogtmann. We also prove Nielsen realisation for limit groups,
and as a byproduct obtain a new proof that limit groups are CAT(). The
proofs rely on a new version of Stallings' theorem on groups with at least two
ends, in which some control over the behaviour of virtual free factors is
gained.Comment: 28 pages, 1 figur
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