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Zariski dense discontinuous surface groups for reductive symmetric spaces

Kazuki Kannaka, Takayuki Okuda, Koichi Tojo

TL;DR

This work advances a broad understanding of how discrete surface groups can act discontinuously on reductive homogeneous spaces, particularly when the stabilizer $H$ is non-compact. It develops a bending/deformation framework for surface groups whose Zariski closure is locally $SL(2,oldsymbol{ m R})$, producing small deformations with Zariski closures equal to even centralizers $G^{ ho}_{ m even}$ and, in symmetric spaces, establishes the existence of Zariski-dense surface subgroups acting properly discontinuously. A key outcome is a criterion based on even nilpotent orbits that guarantees Zariski-dense, non-virtually-abelian, and non-standard actions in many reductive settings, with explicit constructions illustrated in the $SU(p,q)/U(p,q-1)$ family. These results contribute to Clifford–Klein form theory and the spectral-analysis program by showing robust mechanisms to realize Zariski-dense, properly discontinuous surface subgroup actions in broad reductive and symmetric contexts.

Abstract

Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The study of deformations of discontinuous groups for $G/H$ was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group $Γ$ admits a non-standard small deformation as a discontinuous group for $G/H$ if $Γ$ is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we also prove that if $G/H$ is a symmetric space and admits some non virtually abelian discontinuous groups, then $G$ contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on $G/H$. As a key part of our proofs, we show that for a discrete surface subgroup $Γ$ of high genus contained in a reductive group $G$, if the Zariski closure of $Γ$ is locally isomorphic to $SL(2,\mathbb{R})$, then $Γ$ admits a small deformation in $G$ whose Zariski closure is a reductive subgroup of the same real rank as $G$.

Zariski dense discontinuous surface groups for reductive symmetric spaces

TL;DR

This work advances a broad understanding of how discrete surface groups can act discontinuously on reductive homogeneous spaces, particularly when the stabilizer is non-compact. It develops a bending/deformation framework for surface groups whose Zariski closure is locally , producing small deformations with Zariski closures equal to even centralizers and, in symmetric spaces, establishes the existence of Zariski-dense surface subgroups acting properly discontinuously. A key outcome is a criterion based on even nilpotent orbits that guarantees Zariski-dense, non-virtually-abelian, and non-standard actions in many reductive settings, with explicit constructions illustrated in the family. These results contribute to Clifford–Klein form theory and the spectral-analysis program by showing robust mechanisms to realize Zariski-dense, properly discontinuous surface subgroup actions in broad reductive and symmetric contexts.

Abstract

Let be a homogeneous space of reductive type with non-compact . The study of deformations of discontinuous groups for was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group admits a non-standard small deformation as a discontinuous group for if is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to . Furthermore, we also prove that if is a symmetric space and admits some non virtually abelian discontinuous groups, then contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on . As a key part of our proofs, we show that for a discrete surface subgroup of high genus contained in a reductive group , if the Zariski closure of is locally isomorphic to , then admits a small deformation in whose Zariski closure is a reductive subgroup of the same real rank as .
Paper Structure (19 sections, 26 theorems, 65 equations)

This paper contains 19 sections, 26 theorems, 65 equations.

Table of Contents

  1. Introduction and main results
  2. Zariski-dense small deformation of Fuchsian surface groups
  3. Non-standard deformation of a discontinuous group for a non-Riemannian homogeneous space
  4. Zariski-dense discontinuous surface group for a reductive symmetric space
  5. Organization of the paper
  6. Preliminaries
  7. Terminology
  8. Preliminaries for the properness criterion
  9. Deformations of surface subgroups of real reductive groups
  10. Zariski closures of surface subgroups in a real reductive group.
  11. Lemmas for the proof of Theorem \ref{['theorem:deformation-zariski-closure']}
  12. Proof of Theorem \ref{['theorem:deformation-zariski-closure']}
  13. Non-standard small deformations of standard discontinuous surface groups
  14. Zariski-dense discontinuous surface groups for symmetric spaces
  15. Homomorphisms from $SL(2,\mathbb{R})$ to $G$
  16. ...and 4 more sections

Key Result

Theorem 1.6

In the setting of Problem prob:Zariski-dense, if the genus $g$ is sufficiently large, then one can construct a small deformation $\rho'$ of $\rho|_{\Gamma_{g}}$ such that the Zariski closure of $\rho'(\Gamma_{g})$ coincides with $G^{\rho}_{\mathrm{even}}$.

Theorems & Definitions (55)

  • Definition 1.5
  • Theorem 1.6: see also Theorem \ref{['theorem:deformation-zariski-closure']}
  • Corollary 1.7
  • Corollary 1.8
  • Theorem 1.10: see also Theorem \ref{['theorem:strong-non-standard']}
  • Corollary 1.11
  • Theorem 1.13: see also Theorem \ref{['theorem:genZariskidenseSurface_on_symm']}
  • Definition 2.1: Kobayashi Kobayashi96
  • Remark 2.3
  • Corollary 2.4
  • ...and 45 more