Algebraic actions II. Groupoid rigidity

Abstract

We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions rationally embed in each other. For specific example classes arising for instance from toral endomorphisms, actions from rings, or actions from commutative algebra, this mutual embedability can be improved in various ways to obtain surprisingly strong rigidity phenomena. This is witnessed in a particularly striking fashion for actions arising from algebraic number theory: We prove that the groupoids associated with the action of the multiplicative monoid of non-zero elements in a ring of algebraic integers on the additive group of the ring remembers the initial algebraic action up to isomorphism, which in turn remembers the isomorphism class of the ring. This resolves an open problem about isomorphisms of Cartan pairs and leads to a dynamical analogue of the Neukirch--Uchida theorem using topological full groups.