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

TL;DR

This paper shows that uniform approximate lattices in semisimple Lie groups uniquely determine the group's structure, including its $ ext{R}$-rank, extending classical lattice results through ergodic theory and model set techniques.

Contribution

It establishes that approximate lattices as regular model sets can recover the ambient group's $ ext{R}$-rank and structure, generalizing lattice determination results.

Findings

Approximate lattices determine the ambient group's $ ext{R}$-rank.

Existence of a conjugate Cartan subgroup intersecting the approximate lattice as a uniform approximate lattice.

Extension of Mostow's classical lattice result to approximate lattices.

Abstract

Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $\Lambda$ 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 $gCg^{-1} \cap \Lambda^2$ 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.