Compact actions and uniqueness of the group measure space decomposition of II${}_1$ factors

We prove that any II${}_1$ factor $L^{\mathrm{\infty }}(X)⋊\mathrm{\Gamma }$ arising from a compact, free, ergodic, probability measure preserving action of a countable group $\mathrm{\Gamma }$ with positive first ${\mathrm{\ell }}^2$-Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy.