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We investigate the asymptotic structure of (possibly type III) crossedproduct von Neumann algebras $M = B \rtimes \Gamma$ arising from arbitraryactions $\Gamma \curvearrowright B$ of bi-exact discrete groups (e.g. freegroups) on amenable von Neumann algebras. We prove a spectral gap rigidityresult for the central sequence algebra $N' \cap M^\omega$ of any nonamenablevon Neumann subalgebra with normal expectation $N \subset M$. We use thisresult to show that for any strongly ergodic essentially free nonsingularaction $\Gamma \curvearrowright (X, \mu)$ of any bi-exact countable discretegroup on a standard probability space, the corresponding group measure spacefactor ${\rm L}^\infty(X) \rtimes \Gamma$ has no nontrivial central sequence.Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], weconstruct, for every $0 < \lambda \leq 1$, a type III$_\lambda$ stronglyergodic essentially free nonsingular action $\mathbf F_\infty \curvearrowright(X_\lambda, \mu_\lambda)$ of the free group $\mathbf F_\infty$ on a standardprobability space so that the corresponding group measure space typeIII$_\lambda$ factor ${\rm L}^\infty(X_\lambda, \mu_\lambda) \rtimes \mathbfF_\infty$ has no nontrivial central sequence by our main result. In particular,we obtain the first examples of group measure space type III factors with nonontrivial central sequence.

Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence