Uniformly super McDuff \(\hbox {II}_1\) factors

We introduce and study the family of uniformly super McDuff \(\hbox {II}_1\) ${\text{II}}_1$ factors. This family is shown to be closed under elementary equivalence and also coincides with the family of \(\hbox {II}_1\) ${\text{II}}_1$ factors with the Brown property introduced in Atkinson et al. (Adv. Math. 396, 108107, 2022). We show that a certain family of existentially closed factors, the so-called infinitely generic factors, are uniformly super McDuff, thereby improving a recent result of Chifan et al. (Embedding Universality for \(\hbox {II}_1\) ${\text{II}}_1$ Factors with Property (T). arXiv preprint, 2022). We also show that Popa’s family of strongly McDuff \(\hbox {II}_1\) ${\text{II}}_1$ factors are uniformly super McDuff. Lastly, we investigate when finitely generic \(\hbox {II}_1\) ${\text{II}}_1$ factors are uniformly super McDuff.