# Local rigidity of certain actions of solvable groups on the boundaries of rank one symmetric spaces

## TL;DR

This paper investigates the local rigidity of specific solvable group actions on the boundary spheres of rank-one symmetric spaces, establishing rigidity results in quaternionic hyperbolic and Cayley hyperplane cases.

## Contribution

It demonstrates local rigidity for certain solvable subgroup actions on boundary spheres of rank-one symmetric spaces, including quaternionic hyperbolic and Cayley hyperplane cases.

## Key findings

- Actions are locally rigid in quaternionic hyperbolic space
- Actions are locally rigid in Cayley hyperplane
- Rigidity results depend on the type of symmetric space

## 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.