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<G'} and {G'<G}. Furthermore, a certain system of linear equations has integer solutions.

We do not know wether the converse holds.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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