Superrigidity for irreducible lattices and geometric splitting

Abstract

We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem.

Improved version of earlier preprint. Definitions 3, 5 and proof of Theorem 55 modified

References in corpus (2)

• Equivariant embeddings of trees into hyperbolic spaces
• Arithmeticity vs. non-linearity for irreducible lattices

Cited by in corpus (29)

• Simplicity and superrigidity of twin building lattices
• Property (T) and rigidity for actions on Banach spaces
• Isometry groups of non-positively curved spaces: structure theory
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• At infinity of finite-dimensional CAT(0) spaces
• Isometry groups of non-positively curved spaces: discrete subgroups
• Fixed points and amenability in non-positive curvature
• On the bounded cohomology of semi-simple groups, S-arithmetic groups and products
• Group actions on dendrites and curves
• Notes on the Multiplicative Ergodic Theorem
• A lattice in more than two Kac--Moody groups is arithmetic
• On the distortion of twin building lattices
• Uniformly convex metric spaces
• Infinite dimensional non-positively curved symmetric spaces of finite rank
• New simple lattices in products of trees and their projections
• Superrigidity of actions on finite rank median spaces
• Actions of certain arithmetic groups on Gromov hyperbolic spaces
• On the structure and arithmeticity of lattice envelopes
• Geometric density for invariant random subgroups of groups acting on CAT(0) spaces
• Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces
• Product groups acting on manifolds
• Super-Rigidity and non-linearity for lattices in products
• Furstenberg Maps For CAT(0) Targets Of Finite Telescopic Dimension
• Weak topology on CAT(0) spaces
• Actions of Amenable Equivalence Relations on CAT(0) Fields
• Isometric group actions with vanishing rate of escape on CAT(0) spaces
• On a weak topology for Hadamard spaces and its applications
• Milnor-Wood inequalities for products
• Representations of infinite dimension orthogonal groups of quadratic forms with finite index