Assouad-Nagata dimension of connected 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.

21 pages. Main theorem has been extended to connected Lie groups. Added section 6, section 7 and example 4.11

References in corpus (3)

• Hurewicz Theorem for Assouad-Nagata dimension
• Assouad-Nagata dimension of tree-graded spaces
• Assuad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics