Quasi-isometric rigidity for a product of lattices

TL;DR

This paper proves that groups quasi-isometric to a product of a non-uniform rank one lattice and a nilpotent lattice are essentially extensions of these lattices, revealing their geometric rigidity.

Contribution

It establishes quasi-isometric rigidity for such lattice products and characterizes their extensions, including conditions for nilcentral extensions.

Findings

Groups quasi-isometric to the product are extensions of the lattices

Such extensions are nilcentral under certain conditions

The result generalizes known rigidity phenomena

Abstract

We demonstrate quasi-isometric rigidity for the product of a non-uniform rank one lattice and a nilpotent lattice. Specifically, we show that any finitely-generated group quasi-isometric to such a product is, up to finite noise, an extension of a non-uniform rank one lattice by a nilpotent lattice. Furthermore, we show under extra conditions that this extension is nilcentral, a notion which generalizes central extensions to extensions by a nilpotent group.