** 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 leaves attached directly to it, and subtrees of depth 2 with leaves each. Thus the combinatorial pattern is determined by the datum .

**Theorem 1** * *

*
*
- All tree-RAAGs of diameter are commensurable.
- Two tree-RAAGs of diameter 4 are commensurable iff they have the same pattern .

* *

** 1.2. Paths **

**Theorem 2** * Two RAAGs defined by paths of distinct resp. lengths and are commensurable iff and . *

** 1.3. Paths versus trees **

**Theorem 3** * Two RAAGs defined resp. by an -path and a tree of diameter 4 are commensurable iff , , , . *

**2. Method **

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

Note that is a Bass-Serre tree.

**3. Mutual embeddability **

**Theorem 4** * If and are RAAGs, then and . Furthermore, a certain system of linear equations has integer solutions. *

We do not know wether the converse holds.

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