Embedddings between partially commutative groups: Two counterexamples

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 (2013) [12]. 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 C4 or P3. In the process, some new embeddings of surface groups into partially commutative groups emerge.