## Notes of Montserrat Casals-Ruiz Oxford lecture 20-03-2017

Commensurability in RAAGs

Joint work with Kazachkov and Zakharov.

Question. When does quasi-isometry imply commensurability?

1. RAAGs defined by trees

I focus of RAAGs which are simultaneously (irreducible) 3-manifold groups. These coincide with RAAGs defined by trees.

1.1. Small diameter trees

A tree of diameter 4 has a central node, with ${\ell}$ leaves attached directly to it, and ${d_i}$ subtrees of depth 2 with ${i}$ leaves each. Thus the combinatorial pattern is determined by the datum ${M(T)=((k_1,1),(k_2,2),...,(k_j,j) ; \ell)}$.

Theorem 1

1. All tree-RAAGs of diameter ${\leq 3}$ are commensurable.
2. Two tree-RAAGs of diameter 4 are commensurable iff they have the same pattern ${M}$.

1.2. Paths

Theorem 2 Two RAAGs defined by paths of distinct resp. lengths ${m}$ and ${n}$ are commensurable iff ${m=3}$ and ${n=4}$.

1.3. Paths versus trees

Theorem 3 Two RAAGs defined resp. by an ${n}$-path and a tree of diameter 4 are commensurable iff ${n=4k+2}$, ${\ell=2}$, ${k_1=k}$, ${k_2=k+1}$.

2. Method

Study non-abelian centralizers. Take them as vertices of a graph whose edges connect centralizers of commuting elements. This graph ${\Delta}$ turns out to be a commensurability invariant. It is usually not locally finite. The RAAG ${G}$ acts on it by conjugation. The quotient ${\Delta/G}$ is a finite graph, which is again a commensurability invariant.

For trees of diameter 4, commensurability of corresponding RAAGs leads to a system of linear equations satisfied by numbers ${k_i}$, which we solve.

Note that ${\Delta}$ is a Bass-Serre tree.

3. Mutual embeddability

Theorem 4 If ${G}$ and ${G'}$ are RAAGs, then ${G and ${G'. Furthermore, a certain system of linear equations has integer solutions.

We do not know wether the converse holds.