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Determining $\mathbb R$-Rank in Semisimple Lie Groups via uniform approximate Lattice arising as Regular Model Sets

Arunava Mandal, Shashank Vikram Singh

Abstract

Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $Λ$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every non-compact Cartan subgroup $C$ of $G$, there exists $g \in G$ such that the intersection $gCg^{-1} \cap Λ^2$ is non-empty and itself forms a uniform approximate lattice, extending a classical result of Mostow for lattices. The proof relies on a Moore-type ergodicity theorem for the hull of a strong approximate lattice, proved here as a key tool. Moreover, we prove that such approximate lattices determine the $\mathbb{R}$-rank of the ambient group $G$, drawing on ideas from the work of Prasad and Raghunathan on lattices.

Determining $\mathbb R$-Rank in Semisimple Lie Groups via uniform approximate Lattice arising as Regular Model Sets

Abstract

Let be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices arising as regular model sets in determine the ambient group in a strong sense. Specifically, for every non-compact Cartan subgroup of , there exists such that the intersection is non-empty and itself forms a uniform approximate lattice, extending a classical result of Mostow for lattices. The proof relies on a Moore-type ergodicity theorem for the hull of a strong approximate lattice, proved here as a key tool. Moreover, we prove that such approximate lattices determine the -rank of the ambient group , drawing on ideas from the work of Prasad and Raghunathan on lattices.

Paper Structure

This paper contains 14 sections, 21 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Moore’s Ergodicity type theorem for hull of a strong approximate lattice
  3. Strong approximate lattice and regular elements
  4. Strong approximate lattice and non-compact Cartan subgroups
  5. Characterization of $\mathbb R$-rank of linear semisimple group by Uniform approximate Lattice
  6. Preliminaries
  7. Uniform approximate lattices
  8. Chabauty-Fell Topology and Strong Approximate Lattices
  9. Model set
  10. Regular and $\mathbb R$-regular elements
  11. Ergodic action on hull of a strong approximate lattice arising as Regular Model Sets
  12. Strong Approximate Lattices and Regular elements
  13. Uniform Approximate Lattices arising as Regular Model Sets and Non-Compact Cartan Subgroups
  14. Determination of the $\mathbb{R}$-Rank of the Ambient Group by Uniform Approximate Lattices arising as a regular model set

Key Result

Theorem 1.1

Let $G$ be a linear semisimple Lie group without a compact factor and let $\Lambda = P_0(\Gamma, W_0)$ be a regular model set over a cut-and-project scheme $(G,H,\Gamma)$, where $H$ is a linear semisimple Lie group without a compact factor and $\Gamma$ is an irreducible lattice in $G\times H$. Let $

Theorems & Definitions (48)

  • Theorem 1.1: Moore’s Ergodicity-type theorem for hull of a strong approximate lattice
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1: Approximate subgroup
  • Definition 2.2: Uniform approximate lattice
  • Definition 2.3: Approximate lattice
  • Definition 2.4: Strong approximate lattice
  • ...and 38 more