Embedding Universality for II1 Factors with Property (T)

The paper’s Theorem A states exactly the three claims under audit: (i) every separable tracial von Neumann algebra embeds into a II1 factor with property (T); (ii) for every acylindrically hyperbolic group H there is a II1 factor Q = W*(π(H)) containing M (hence if H has property (T), then so does Q); and (iii) one can choose a property (T) II1 factor P with Out(P) = {e} and F(P) = {1} containing M. These appear verbatim in the introduction and are proved via the wreath‑like product framework and an extension-of-representations construction (Proposition 3.8), together with CIOS results (Theorem 2.2) and a rigidity theorem (Theorem 4.1) that yields trivial outer automorphism and fundamental groups for the constructed factors. See Theorem A and its proof sketch and details, Proposition 3.8, Theorem 2.2, and Section 5.3 for the Out/F conclusions . The candidate solution follows the same mechanism and cites the same ingredients. The only minor imprecision is the opening summary’s suggestion that the Q from (ii) can also be arranged to have Out(Q) = {e} and F(Q) = {1}; in the paper, that strengthened conclusion is provided separately by (iii), without insisting that this P arise from an arbitrary prescribed H. Apart from that, the model’s outline aligns with the paper’s construction, including the step that if H has property (T), then Q has property (T), which is exactly how the paper concludes part (1) of Theorem A .