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Using an extension of techniques of Ozawa and Popa, we give an example of anon-amenable strongly solid $\rm{II}_1$ factor $M$ containing an "exotic"maximal abelian subalgebra $A$: as an $A$,$A$-bimodule, $L^2(M)$ is neithercoarse nor discrete. Thus we show that there exist $\rm{II}_1$ factors withsuch property but without Cartan subalgebras. It also follows from Voiculescu'sfree entropy results that $M$ is not an interpolated free group factor, yet itis strongly solid and has both the Haagerup property and the complete metricapproximation property.

Strongly solid ${\rm II_1}$ factors with an exotic MASA