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Kulkarni limit sets for cyclic quaternionic projective groups

Sandipan Dutta, Krishnendu Gongopadhyay, Rahul Mondal

TL;DR

This work extends the theory of Kulkarni limit sets to cyclic subgroups of ${\rm PSL}(n+1,\mathbb{H})$ acting on quaternionic projective space $\mathbb{P}_{\mathbb{H}}^n$, generalizing prior complex and low-dimensional quaternionic results. By leveraging lifts to ${\rm SL}(n+1,\mathbb{H})$ and the framework of pseudo-projective transformations, the authors classify group elements via quaternionic Jordan forms and compute the corresponding Kulkarni limit sets for elliptic, parabolic, loxodromic, and loxoparabolic cyclic subgroups. The main contributions include a detailed case-by-case determination of $\Lambda_{Kul}(G)$ for each conjugacy type, a unifying methodology across dimensions, and a quaternionic-specific account of phenomena that differ from the complex case due to non-commutativity. The results, summarized in a comprehensive table, highlight how the limit-set structure depends on diagonal and Jordan blocks and demonstrate both alignments and deviations from complex projective dynamics, with potential implications for higher-rank quaternionic Kleinian-type groups.

Abstract

We consider the natural action of the quaternionic projective linear group $\mathrm{PSL}(n+1,\mathbb{H})$ on the quaternionic projective space $\mathbb{P}^n_{\mathbb{H}}$. We compute the Kulkarni limit sets for the cyclic subgroups of $\mathrm{PSL}(n+1,\mathbb{H})$.

Kulkarni limit sets for cyclic quaternionic projective groups

TL;DR

This work extends the theory of Kulkarni limit sets to cyclic subgroups of acting on quaternionic projective space , generalizing prior complex and low-dimensional quaternionic results. By leveraging lifts to and the framework of pseudo-projective transformations, the authors classify group elements via quaternionic Jordan forms and compute the corresponding Kulkarni limit sets for elliptic, parabolic, loxodromic, and loxoparabolic cyclic subgroups. The main contributions include a detailed case-by-case determination of for each conjugacy type, a unifying methodology across dimensions, and a quaternionic-specific account of phenomena that differ from the complex case due to non-commutativity. The results, summarized in a comprehensive table, highlight how the limit-set structure depends on diagonal and Jordan blocks and demonstrate both alignments and deviations from complex projective dynamics, with potential implications for higher-rank quaternionic Kleinian-type groups.

Abstract

We consider the natural action of the quaternionic projective linear group on the quaternionic projective space . We compute the Kulkarni limit sets for the cyclic subgroups of .
Paper Structure (14 sections, 16 theorems, 66 equations, 1 table)

This paper contains 14 sections, 16 theorems, 66 equations, 1 table.

Table of Contents

  1. Introduction
  2. Structure of the paper:
  3. Notations
  4. Preliminaries
  5. The Quaternion Projective Space $\mathbb P^n_\mathbb {H}$
  6. Projective Transformations
  7. Pseudo-projective Transformations
  8. Grassmanian
  9. Some useful lemmas
  10. Dual Projective Space
  11. Elliptic case
  12. Parabolic case
  13. Loxodromic case
  14. Loxoparabolic case

Key Result

Proposition 1.3

Let $X$ be a locally compact Hausdorff space and $G$ be a group acting on $X$, then $G$ acts properly discontinuously on $\Omega(G)$. In addition, $\Omega(G)$ is an open subset of $X$. Further, if $\Omega(G)\neq \emptyset$, then $G$ is discrete.

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3: cf. KUL
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: cf. he
  • Lemma 2.4: Jordan form in $\mathrm{M}(n,\mathbb {H})$, cf. rodman
  • Definition 2.5
  • Lemma 2.6
  • ...and 23 more