Montserrat Casals-Ruiz: Embedability between partially commutative groups

Abstract: Partially commutative groups (also known as right-angled Artin groups) appear naturally in many branches of mathematics such as geometry, algebra and computer sciences. Many recent results exhibit the potential of this class of groups, show that several outstanding problems are closely related to them and make them of high interest in group theory and low-dimensional topology. An illustration of this is the recent work of Dani Wise in which he shows that many groups are virtually subgroups of partially commutative groups and uses this fact to solve some well-known problems such as Baumslag's conjecture on residual finiteness of one-relator groups with torsion and the virtually fibered conjecture for Haken hyperbolic 3-manifolds.

A very natural question for this class of groups is to determine when a given partially commutative group is a subgroup of another one. In their recent preprint, Kim and Koberda give a necessary condition for the embedability between partially commutative groups and conjecture a criterion. In this talk, I will give counterexamples to the conjectured criterion and to the Weakly-Chordal conjecture. I will further discuss a criterion for embedability and its applications to the study of universal equivalence and quasi-isometry of partially commutative groups.