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.
Cuntz, C*-algebras associated with the ax + b-semigroup over N, K-theory and noncommutative geometry, 201– 215, EMS Ser
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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.