W∗-Rigidity Paradigms for Embeddings of II1 Factors

We undertake a systematic study of W ∗ -rigidity paradigms for the embeddability relation ↪ between separable II 1 factors and its stable version ↪ 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 ↦ H G from the category of groups into icc groups, so that L ( H G 1 ) ↪ s L ( H G 2 ) iff G 1 ↪ G 2 . We construct complete intervals of II 1 factors, including a strict chain of II 1 factors ( M k ) k ∈ Z with the property that if N is any II 1 factor with M i ↪ s N and N ↪ s M j , then N ≅ M k t for some i ≤ k ≤ j and t > 0 .