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

We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras $M=B⋊\mathrm{\Gamma }$ arising from arbitrary actions $\mathrm{\Gamma }\curvearrowright B$ 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 $N^{\mathrm{\prime }}\cap M^{\omega }$ 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 $\mathrm{\Gamma }\curvearrowright (X,\mu )$ of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor ${\mathrm{L}}^{\mathrm{\infty }}(X)⋊\mathrm{\Gamma }$ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every $0<\lambda \le 1$, a type III${}_{\lambda }$ strongly ergodic essentially free nonsingular action $\mathbf F_\infty \curvearrowright (X_\lambda, \mu_\lambda)$ofthefreegroup$\mathbf F_\infty$ 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.