Publication

Property (T) and rigidity for actions on Banach spaces

Nicolas Monod
2007
Journal paper

Abstract

We study property (T) and the fixed point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that in fact it is independent of 1 ≤ p < ∞. We show that the fixed point property for Lp follows from property (T) when 1 < p < 2+ε. For simple Lie groups and their lattices, we prove that the fixed point property for Lp holds for any 1 < p < ∞ 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.

Official source

https://infoscience.epfl.ch/record/128764?ln=en

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Ontological neighbourhood

Mathematics

Algebra: Representation theory

Analysis: Functional analysis

Related concepts (43)

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

Simple Lie group

In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension.

Algebraic group

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.

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Related MOOCs (9)

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Property (T) and rigidity for actions on Banach spaces

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