Embeddings between partially commutative groups: two counterexamples

TL;DR

This paper presents two counterexamples to prominent conjectures in the theory of partially commutative groups and extends known classes where the Extension Graph Conjecture holds, revealing new embeddings of surface groups.

Contribution

It provides counterexamples to the Extension Graph and Weakly Chordal Conjectures and extends the class of groups satisfying the Extension Graph Conjecture.

Findings

Counterexamples to the Extension Graph and Weakly Chordal Conjectures.

Extension of the class of groups where the Extension Graph Conjecture holds.

Discovery of new embeddings of surface groups into partially commutative groups.

Abstract

In this note we give two examples of partially commutative subgroups of partially commutative groups. Our examples are counterexamples to the Extension Graph Conjecture and to the Weakly Chordal Conjecture of Kim and Koberda, \cite{KK}. On the other hand we extend the class of partially commutative groups for which it is known that the Extension Graph Conjecture holds, to include those with commutation graph containing no induced $C_4$ or $P_3$. In the process, some new embeddings of surface groups into partially commutative groups emerge.