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Title: Measure equivalence rigidity of 2-dimensional Artin groups of hyperbolic type.

Abstract: The notion of measure equivalence between countable groups was introduced by Gromov as a measure-theoretic analogue of quasi-isometry. We study the class of 2-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and show that if two groups from this class are measure equivalent, then their "curve graphs" are isomorphic. This reduces the question of measure equivalence of these groups to a combinatorial rigidity question concerning their curve graphs; in particular, we deduce measure equivalence superrigidity results for a class of Artin groups whose curve graphs are known to be rigid from a previous work of Crisp. There are two main ingredients in the proof of independent interest. The first is a more general result concerning boundary amenability of groups acting on certain CAT(-1) spaces. The second is a structural similarity between these Artin groups and mapping class groups from the viewpoint of measure equivalence. This is joint work with Camille Horbez.

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