High Rank and Multiplicity in Random and Perfect Profinite Groups

Source abstract

We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu--Wood and Sawin--Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu--Wood and Sawin--Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with $d$ generators and $d$ relations, and with no fewer relations on $d$ generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.