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We prove that for a fairly large family of HNN extensions \Gamma, the groupmeasure space II_1 factor L^\infty(X) \rtimes \Gamma given by an arbitrary freeergodic probability measure preserving action of \Gamma, has a unique groupmeasure space Cartan subalgebra up to unitary conjugacy. We deduce from thisnew examples of W^*-superrigid group actions, i.e. where the II_1 factorL^\infty(X) \rtimes \Gamma entirely remembers the group action that it wasconstructed from.

HNN extensions and unique group measure space decomposition of II$_{1}$ factors