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