Almost homogeneous manifolds with boundary

TL;DR

This paper investigates the classification of Lie group actions on manifolds with boundary, showing that many such actions, including those related to negatively curved symmetric spaces, have infinitely many topologically conjugate variants.

Contribution

It demonstrates that the set of actions topologically conjugate to a given transitive action on a manifold with boundary can be infinite, especially in the context of compactifications of negatively curved symmetric spaces.

Findings

The set of conjugate actions is often infinite.

Includes cases of compactifications of negatively curved symmetric spaces.

Provides insights into the structure of group actions on manifolds with boundary.

Abstract

Let $\rho_0$ be an action of a Lie group on a manifold with boundary that is transitive on the interior. We study the set of actions that are topologically conjugate to $\rho_0$, up to smooth or analytic change of coordinates. We show that in many cases, including the compactifications of negatively curved symmetric spaces, this set is infinite.