Infinite generation of non-cocompact lattices on right-angled buildings

TL;DR

This paper proves that non-cocompact lattices with a strict fundamental domain acting on right-angled buildings are infinitely generated, using properties of subcomplexes called tree-walls.

Contribution

It establishes that such lattices are not finitely generated if they possess a strict fundamental domain, advancing understanding of lattice structures in right-angled buildings.

Findings

Non-cocompact lattices with strict fundamental domains are infinitely generated.

Tree-wall subcomplexes are key in proving infinite generation.

Provides new insights into the structure of lattices in right-angled buildings.

Abstract

Let \Gamma be a non-cocompact lattice on a locally finite regular right-angled building X. We prove that if \Gamma has a strict fundamental domain then \Gamma is not finitely generated. We use the separation properties of subcomplexes of X called tree-walls.