In this talk, we recall Borel reducibility among equivalence relations first. Then we introduce equivalence relations $E(G)$ induced by Polish groups $G$. Main part of this talk is to precent many rigid results concerning various kinds of Polish groups: non-archimedean, TSI, CLI, $\alpha$-unbalanced, abelian, locally compact, Lie groups, and Banach spaces (as additive groups) and so on.

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