Boundary and Rigidity of Nonsingular Bernoulli Actions

Let $ G be a countable discrete group and consider nonsingular Bernoulli shift action \curvearrowright \prod_{g\in }(\{0,1\},\mu_g)$ with two base points. When is exact, under certain finiteness assumption on the measures $\{\mu_g\}_{g\in }$, we construct boundary for crossed product C$^*$-algebra that admits some commutativity amenability in sense of Ozawa's bi-exactness. As consequence, obtain any such solid. This generalizes solidity measure preserving actions by Ozawa Chifan--Ioana, first rigidity result non case. For proof, use anti-symmetric Fock spaces left creation operators to therefore having points crucial.