Indiscrete Common Commensurators
Jingyin Huang, Mahan Mj

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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology
