W*-rigidity paradigms for embeddings of II$_1$ factors

We undertake a systematic study of W*-rigidity paradigms for the embeddability relation $\hookrightarrow$ between separable II$_1$ factors and its stable version $\hookrightarrow_s$, obtaining large families of non stably isomorphic II$_1$ factors that are mutually embeddable and families of II$_1$ factors that are mutually non stably embeddable. We provide an augmentation functor $G \mapsto H_G$ from the category of groups into icc groups, so that $L(H_{G_1}) \hookrightarrow_s L(H_{G_2})$ iff $G_1 \hookrightarrow G_2$. We construct complete intervals of II$_1$ factors, including a strict chain of II$_1$ factors $(M_k)_{k\in \mathbb{Z}}$ with the property that if $N$ is any II$_1$ factor with $M_i \hookrightarrow_s N$ and $N \hookrightarrow_s M_{j}$, then $N \cong M_k^t$ for some $i \leq k \leq j$ and $t > 0$.