Indiscrete Common Commensurators

TL;DR

This paper investigates the properties of common commensurators of discrete subgroups in CAT(0) space lattices, revealing that certain hypotheses about their discreteness do not hold universally, with specific examples analyzed.

Contribution

It introduces a new framework for understanding common commensurators in CAT(0) spaces and demonstrates the failure of the Greenberg-Shalom hypothesis in this setting.

Findings

Greenberg-Shalom hypothesis fails in generality

Examples by Burger and Mozes have discrete common commensurators

Discreteness of common commensurators is not guaranteed under strong conditions

Abstract

We develop a framework for common commensurators of discrete subgroups of lattices in isometry groups of CAT(0) spaces. We show that the Greenberg-Shalom hypothesis about discreteness of common commensurators of Zariski dense subgroups and lattices fails in this generality, even if one imposes strong finiteness conditions. We analyze some examples due to Burger and Mozes in this context and show that they have discrete common commensurator.