Groupoids arising from algebraic actions of semigroups are rigid: isomorphisms recover rational embeddings of the actions, and for rings of algebraic integers the groupoid determines the ring up to isomorphism.
Matui, Homology and topological full groups of ´ etale groupoids on totally disconnected spaces, Proc
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Algebraic actions II. Groupoid rigidity
Groupoids arising from algebraic actions of semigroups are rigid: isomorphisms recover rational embeddings of the actions, and for rings of algebraic integers the groupoid determines the ring up to isomorphism.