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