Jingyin Huang: Integral measure equivalence versus quasiisometry for some right-angled Artin groups.

Atelier sur complexes cubiques et géométrie combinatoire/ Workshop on Cube Complexes and Combinatorial Geometry (Juin/June 19-30 2023)

Juin/June 26 :

Two finitely generated groups G and H are quasi-isometric (QI), if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood. We will start with a discussion of an interesting similarity between some QI invariants and ME invariants for general cubical groups. Then we look at the case of right-angled Artin groups. We show that if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. This is perhaps the first instance a rigidity result in the ME side is obtained via establishing quasi-isometry. This is joint work with Camille Horbez.