Abstract
Let $ G $ be the group of orientation-preserving isometries of a rank-one symmetric space $ X $ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $ \Gamma \subset G $ on the boundary of $ X $, which is diffeomorphic to a sphere. When $ X $ is a quaternionic hyperbolic space or the Cayley hyperplane, the action we constructed is locally rigid.
Highlights
One of the most active areas of the study of rigidity of group actions is around the Zimmer program, in which many remarkable properties of actions of a lattice Γ of a higher rank Lie group have been discovered
Let G be the group of orientation-preserving isometries of a rank one symmetric space of non-compact type, Γ a standard subgroup of G, and l|Γ the action of Γ on G/P by left translations
Let G be the group of orientation-preserving isometries of a rank one symmetric space of non-compact type, Γ a standard subgroup of G, and l : P → J3(G/P, o) the homomorphism into the group of 3-jets at o ∈ G/P induced by the action of P on G/P by left translations
Summary
One of the most active areas of the study of rigidity of group actions is around the Zimmer program, in which many remarkable properties of actions of a lattice Γ of a higher rank Lie group have been discovered. The main theorem is reduced to local rigidity of a homomorphism of Γ into the group G(G/P, o) of germs of diffeomorphisms defined around o ∈ G/P and fixing o ∈ G/P. Such a problem of classification of “local action” around a fixed point can be found in [9].
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