Cohomology of hyperfinite Borel actions - Bezuglyi, Sergey I.; Sanadhya, Shrey (Groups, geometry and dynamics, 2021)
We study cocycles of countable groups Γ of Borel automorphisms of a standard Borel space (X, \mathcal{B}) taking values in a locally compact second countable group G . We prove that for a hyperfinite group Γ the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the 2 -odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup H of G . We also provide a Borel version of Gottschalk–Hedlund theorem.