Groups acting on manifolds: around the Zimmer program

Abstract

This paper is a survey on the {\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are $(1)$ to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, $(2)$ to indicate the current state of the art on as many of these conjectures and questions as possible and $(3)$ to indicate a wide variety of open problems and directions of research.

Survey article, 84 pages, almost final version, comments welcome but only if made quickly

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• Bounded cohomology of subgroups of mapping class groups
• Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions
• Distortion in transformation groups
• Mapping Class Groups do not have Kazhdan's Property (T)
• Area preserving group actions on surfaces
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Cited by in corpus (30)

• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• A survey of Measured Group Theory
• Continuity of discrete homomorphisms of diffeomorphism groups
• Finite group actions on manifolds without odd cohomology
• Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds
• Euler characteristics and actions of automorphism groups of free groups
• Quasihomogeneous analytic affine connections on surfaces
• Rigidity and flexibility of group actions on the circle
• Holomorphic actions of higher rank lattices in dimension three
• Jordan groups and automorphism groups of algebraic varieties
• Finite group actions on 4-manifolds with nonzero Euler characteristic
• Non left-orderability of lattices in higher rank semi-simple Lie groups
• Cohomological obstructions to Nielsen realization
• The Burnside problem for ${\text{Diff}}_{\text{Vol}}({\mathbb{S}}^2)$
• Global smooth and topological rigidity of hyperbolic lattice actions
• Symmetric random walks on Homeo+(R)
• Finite groups acting symplectically on $T^2\times S^2$
• Matrix group actions on product of spheres and Zimmer's program
• Smooth Factors of Projective Actions of Higher Rank Lattices and Rigidity
• Superrigidity, arithmeticity, normal subgroups: results, ramifications and directions
• Piecewise linear actions and Zimmer's program
• Zimmer's conjecture for lattice actions: the $\mathrm{SL}(n,\mathbb{C})$-case
• Special $p$-groups acting on compact manifolds
• One some planar Baumslag-Solitar actions
• A survey of topological Zimmer's program
• Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo-)Anosov elements
• Global rigidity of actions by higher rank lattices with dominated splitting
• Volume-preserving actions of simple algebraic Q-groups on low-dimensional manifolds
• Actions of arithmetic groups on homology spheres and acyclic homology manifolds
• Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groups