Research on discrete groups under measure equivalence

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We introduced coupling rigidity for a discrete countable group, and solved the problem when the amalgamated free product of two discrete countable groups having rigidity has rigidity. As its consequence, we constructed new groups having rigidity, and showed that those groups satisfy rigidity in a sense of orbit equivalence. We computed the abstract commensurators of the Torelli group, the Johnson kernel and the surface braid group, which are special subgroups of the mapping class group of a surface. In particular, we succeeded in describing any automorphism of those groups.