Irreducible lattices, invariant means, and commensurating actions

SEMIGROUP ACTIONS , WEAK ALMOST PERIODICITY, AND INVARIANT MEANS

Let S be a topological semigroup acting on a topological space X. We develop the theory of (weakly) almost periodic functions on X, with respect to S, and form the (weakly) almost periodic compactifications of X and S, with respect to each other. We then consider the notion of an action of Son a Banach space, and on its dual, and after defining S-invariant means for such a space, we give a...

Amenable Actions, Invariant Means and Bounded Cohomology

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a poi...

Rigidity of Commensurators and Irreducible Lattices

Superrigidity for Irreducible Lattices and Geometric Splitting

1.1. Superrigidity. In the early seventies, Margulis proved his celebrated superrigidity theorem for irreducible lattices in semisimple Lie and algebraic groups of higher rank. One of the motivations for this result is that it implies arithmeticity : a complete classification of higher rank lattices. In the case where the semisimple group is not almost simple, superrigidity reads as follows (se...

Irreducible elements in multi-adjoint concept lattices

One of the most important elements in a lattice are the irreducible elements. For example, when the lattice is finite, which is usual in the computational case, it forms a base from which the complete lattice is obtained. These elements are also important in Formal Concept Analysis, since they are the basic information of a relational system. In the general fuzzy framework of multi-adjoint conc...