# On the distortion of twin building lattices

## TL;DR

This paper proves that twin building lattices are undistorted in their ambient groups, establishing their quasi-isometric embedding and implications for the diversity of finitely presented simple groups.

## Contribution

It demonstrates the undistorted nature of twin building lattices and links this property to the quasi-flat rank and classification of simple groups.

## Key findings

- Twin building lattices are undistorted in their ambient groups.
- The orbit map of the lattice to twin buildings is a quasi-isometric embedding.
- There are infinitely many quasi-isometry classes of finitely presented simple groups.

## Abstract

We show that twin building lattices are undistorted in their ambient group; equivalently, the orbit map of the lattice to the product of the associated twin buildings is a quasi-isometric embedding. As a consequence, we provide an estimate of the quasi-flat rank of these lattices, which implies that there are infinitely many quasi-isometry classes of finitely presented simple groups. In an appendix, we describe how non-distortion of lattices is related to the integrability of the structural cocycle.