Factors of type II$_1$ without non-trivial finite index subfactors

We call a subfactor $N \subset M$ trivial if it is isomorphic with the obvious inclusion of $N$ in $\operatorname{M}_n(\mathbb{C}) \otimes N$. We prove the existence of type II$_1$ factors $M$ without non-trivial finite index subfactors. Equivalently, every $M$-$M$-bimodule with finite coupling constant, both as a left and as a right $M$-module, is a multiple of $L^2(M)$. Our results rely on the recent work of Ioana, Peterson and Popa, who proved the existence of type II$_1$ factors without outer automorphisms.