Seminar

Group Theory Seminar

Hopfian wreath products and the stable finiteness conjecture

Speaker:  Francesco Fournier-Facio (ETH Zurich)
Date:  Tuesday, 18 April 2023 - 11:30
Place:  Aula 520, Módulo 17, Departamento de Matemáticas, UAM

Abstract:

Wreath products appear often as a case study in questions related to residual finiteness, thanks to a beautiful and simple characterization of Gruenberg. A related property is the Hopf property: a group is Hopfian if every self-epimorphism is an isomorphism. Every finitely generated residually finite group is Hopfian, which motivates looking at the Hopf property for wreath products, in hope of a simple characterization analogous to Gruenberg's.
It turns out that this problem is infinitely harder than Gruenberg's, even when focusing on the following special case: if G is finitely generated abelian, and H is finitely generated Hopfian, is G \wr H Hopfian? We will see that this question is equivalent to one of the most longstanding open problems in group theory: Kaplansky's stable finiteness conjecture, which is strongly related to the zero-divisor and idempotent conjectures, to the existence of a non-sofic group, and to Gottschalk's surjunctivity conjecture.
This is joint work with Henry Bradford (Cambridge).