Cohomogeneity one actions on symmetric spaces of noncompact type and higher rank

TL;DR

This paper develops a structural framework for classifying cohomogeneity one actions on symmetric spaces of noncompact type, including reducible and higher rank cases, and applies it to specific spaces like SL(n,R)/SO(n).

Contribution

It introduces a new structural result for cohomogeneity one actions and reduces the classification problem on reducible spaces to their irreducible components.

Findings

Classified cohomogeneity one actions on SL(n,R)/SO(n) for n>1.

Reduced classification on reducible spaces to irreducible factors.

Extended classification to finite products of hyperbolic spaces.

Abstract

We develop a new structural result for cohomogeneity one actions on (not necessarily irreducible) symmetric spaces of noncompact type and arbitrary rank. We apply this result to classify cohomogeneity one actions on SL(n,R)/SO(n), n>1, up to orbit equivalence. We also reduce the classification problem on a reducible space to the classification on each one of its irreducible factors, which in particular allows to classify cohomogeneity one actions on any finite product of hyperbolic spaces.