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Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals. More formally does ...

For the purpose of this question, a group is amenable iff there exists a Følner sequence. Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...

Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...