Assouad-Nagata dimension of connected Lie groups

TL;DR

This paper establishes that the asymptotic Assouad-Nagata dimension of connected Lie groups matches their topological dimension after factoring out a maximal compact subgroup, with implications for polycyclic groups and lattice quasi-isometries.

Contribution

It computes the Assouad-Nagata dimension for connected solvable and semisimple Lie groups and relates it to topological and algebraic invariants, providing new insights into geometric group theory.

Findings

Asymptotic Assouad-Nagata dimension equals topological dimension of G/C.

Polycyclic groups' asymptotic Assouad-Nagata dimension equals their Hirsch length.

Certain wreath-type groups cannot be quasi-isometric to cocompact lattices in Lie groups.

Abstract

We prove that the asymptotic Assouad-Nagata dimension of a connected Lie group $G$ equipped with a left-invariant Riemannian metric coincides with its topological dimension of $G/C$ where $C$ is a maximal compact subgroup. To prove it we will compute the Assouad-Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad-Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometric to any cocompact lattice on a connected Lie group.