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Classification of cohomogeneity-one actions on symmetric spaces of noncompact type

Ivan Solonenko, Víctor Sanmartín-López

TL;DR

This work resolves the full classification problem for isometric cohomogeneity-one actions on symmetric spaces of noncompact type up to orbit equivalence. It combines three pillars—horospherical and solvable constructions, canonical extensions from boundary components, and the nilpotent-construction method—through a detailed analysis of transitive subgroups, root-system constraints, and the solvable model $AN$. Central to the achievement is the admissibility-and-protohomogeneity framework, which, together with root-vector elimination and boundary-component reduction, reduces the nilpotent-construction problem to a finite set of possibilities; in rank>1, these yield either canonical extensions or totally geodesic singular orbits. The main theorem enumerates the action types (horospherical $H_\ell$, simple-root $H_{\alpha_i}$, canonical extensions from boundary components, diagonal reducible-boundary actions, and $H_{j,\mathfrak{w}}$-type nilpotent-construction actions) and asserts that every C1-action on $M$ is orbit-equivalent to one of these. The results provide a complete structural framework for understanding homogeneous hypersurfaces and related CPC submanifolds in noncompact symmetric spaces, with potential implications for geometric flows and foliations on these spaces.

Abstract

We complete the classification of isometric cohomogeneity-one actions on all symmetric spaces of noncompact type up to orbit equivalence.

Classification of cohomogeneity-one actions on symmetric spaces of noncompact type

TL;DR

This work resolves the full classification problem for isometric cohomogeneity-one actions on symmetric spaces of noncompact type up to orbit equivalence. It combines three pillars—horospherical and solvable constructions, canonical extensions from boundary components, and the nilpotent-construction method—through a detailed analysis of transitive subgroups, root-system constraints, and the solvable model . Central to the achievement is the admissibility-and-protohomogeneity framework, which, together with root-vector elimination and boundary-component reduction, reduces the nilpotent-construction problem to a finite set of possibilities; in rank>1, these yield either canonical extensions or totally geodesic singular orbits. The main theorem enumerates the action types (horospherical , simple-root , canonical extensions from boundary components, diagonal reducible-boundary actions, and -type nilpotent-construction actions) and asserts that every C1-action on is orbit-equivalent to one of these. The results provide a complete structural framework for understanding homogeneous hypersurfaces and related CPC submanifolds in noncompact symmetric spaces, with potential implications for geometric flows and foliations on these spaces.

Abstract

We complete the classification of isometric cohomogeneity-one actions on all symmetric spaces of noncompact type up to orbit equivalence.
Paper Structure (14 sections, 46 theorems, 61 equations, 6 figures)

This paper contains 14 sections, 46 theorems, 61 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric spaces of noncompact type
  4. Parabolic subalgebras and subgroups
  5. The extrinsic geometry of singular orbits
  6. Cohomogeneity-one actions
  7. Types of cohomogeneity-one actions
  8. The current state of classification
  9. The admissibility condition
  10. Transitive groups of isometries of symmetric spaces of noncompact type
  11. The redundancy
  12. Root vector elimination
  13. Totally geodesic submanifolds and the singular orbit
  14. Solving the nilpotent construction problem

Key Result

Theorem A

In the special cases of $\mathbb{R} H^n$ and $\mathbb{C} H^n$, a similar result was obtained in transitive_CH^n Let $M = G/K$ be a symmetric space of noncompact type and $H \subseteq G$ a Lie subgroup acting on $M$ transitively. Then there exist an Iwasawa decomposition $\mathfrak{{g}} = \mathfrak{{

Figures (6)

  • Figure 1: $\Updelta_j^1$ for $\Upsigma \cong \mathrm{{F}}_4$ and $j = 1$.
  • Figure 2: $\Updelta_j^1$ for $\Upsigma \cong \mathrm{{F}}_4$ and $j = 4$.
  • Figure 3: $\Updelta_j^1$ for $\Upsigma \cong \mathrm{{C}}_5$ and $j = 1$.
  • Figure 4: $\Updelta_j^1$ for $\Upsigma \cong \mathrm{{C}}_5$ and $j = 5$.
  • Figure 5: $\Updelta_j^1$ when $\Upsigma \cong \mathrm{{B}}_5$ or $\mathrm{{BC}}_5$ and $j = 1$.
  • ...and 1 more figures

Theorems & Definitions (83)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 73 more