1,863 research outputs found
Simple groups without lattices
We show that the group of almost automorphisms of a d-regular tree does not
admit lattices. As far as we know this is the first such example among
(compactly generated) simple locally compact groups.Comment: 17 pages. Revised according to referee's repor
Ideal bicombings for hyperbolic groups and applications
For every hyperbolic group and more general hyperbolic graphs, we construct
an equivariant ideal bicombing: this is a homological analogue of the geodesic
flow on negatively curved manifolds. We then construct a cohomological
invariant which implies that several Measure Equivalence and Orbit Equivalence
rigidity results established by Monod-Shalom hold for all non-elementary
hyperbolic groups and their non-elementary subgroups. We also derive
superrigidity results for actions of general irreducible lattices on a large
class of hyperbolic metric spaces.Comment: Substantial generalizeation; now the results hold for a general class
of hyperbolic metric spaces (rather than just hyperbolic groups
Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces
We define a Toledo number for actions of surface groups and complex
hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which
allows us to define maximal representations. When the target is not of tube
type we show that there cannot be Zariski-dense maximal representations, and
whenever the existence of a boundary map can be guaranteed, the representation
preserves a finite dimensional totally geodesic subspace on which the action is
maximal. In the opposite direction we construct examples of geometrically dense
maximal representation in the infinite dimensional Hermitian symmetric space of
tube type and finite rank. Our approach is based on the study of boundary maps,
that we are able to construct in low ranks or under some suitable
Zariski-density assumption, circumventing the lack of local compactness in the
infinite dimensional setting.Comment: Comments are welcome! The maximality assumption was unfortunately
missing in Theorem 1.1 and 1.4 of the first versio
Finite and infinite quotients of discrete and indiscrete groups
These notes are devoted to lattices in products of trees and related topics.
They provide an introduction to the construction, by M. Burger and S. Mozes, of
examples of such lattices that are simple as abstract groups. Two features of
that construction are emphasized: the relevance of non-discrete locally compact
groups, and the two-step strategy in the proof of simplicity, addressing
separately, and with completely different methods, the existence of finite and
infinite quotients. A brief history of the quest for finitely generated and
finitely presented infinite simple groups is also sketched. A comparison with
Margulis' proof of Kneser's simplicity conjecture is discussed, and the
relevance of the Classification of the Finite Simple Groups is pointed out. A
final chapter is devoted to finite and infinite quotients of hyperbolic groups
and their relation to the asymptotic properties of the finite simple groups.
Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups
updated; more examples added in Section 4; new Proposition 5.1
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