We prove that the group C ∼ $\widetilde{\mathcal {C}}$ of equivariant concordance directed strongly invertible knots, defined by Sakuma (On Algebraic and topological theories (Kinosaki, 1984), Kinokuniya Company Ltd., Tokyo, 1986, pp. 176–196), is not abelian. do so exhibiting an infinite family non-trivial commutators.

We show that a Borel action of standard group which is isomorphic to sum countable abelian with real lines and circles induces an orbit equivalence relation hypersmooth, i.e., reducible eventual agreement on sequences reals, it follows from this result along the structure theory for locally compact groups actions Polish LCA induce relations are essentially hyperfinite, extending Gao Jackson ans...