Local rigidity of higher rank lattice actions on flag manifolds - Part 1

By Thang Nguyen

Appears in collection : 2024 - T2 - Group actions and rigidity: around the Zimmer program

The main focus of the minicourse is to study local rigidity of standard actions of uniform lattices in higher rank Lie groups on flag manifolds within the space of actions by homeomorphisms. The rigidity in more differentiable actions often relies on the stability of uniform hyperbolic systems. In the setting of actions by homeomorphisms, We explore a much weaker form of stability derived from coarse geometry. We will go over how dynamics and coarse geometry intertwine to achieve a desired local rigidity. At the end of the minicourse, we will extend our discussion to other natural boundary actions.

The outlined of topics in the minicourse include:

• Local rigidity: problems and results.

• Stability of uniform hyperbolic systems and uniformly transverse foliations.

• Geometry and coarse geometry of symmetric spaces.

• Weyl chamber flows.

• Suspension and developing.

• Barycenter and its regularity.

• Proof of local rigidity for actions on flag manifolds.

• Actions on geodesic boundaries and some related actions.

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