Hopfian wreath products and the stable finiteness conjecture

We study the Hopf property for wreath products of finitely generated groups, focusing on the case of an abelian base group. Our main result establishes a strong connection between this problem and Kaplansky’s stable finiteness conjecture. Namely, the latter holds true if and only if for every finitely generated abelian group A and every finitely generated Hopfian group \(\Gamma \) $\mathrm{\Gamma }$ the wreath product \(A \wr \Gamma \) $A\wr \mathrm{\Gamma }$ is Hopfian. In fact, we characterize precisely when \(A \wr \Gamma \) $A\wr \mathrm{\Gamma }$ is Hopfian, in terms of the existence of one-sided units in certain matrix algebras over \({\mathbb {F}}_p[\Gamma ]\) ${\mathbb{F}}_p[\mathrm{\Gamma }]$ , for every prime p occurring as the order of some element in A. A tool in our arguments is the fact that fields of positive characteristic locally embed into matrix algebras over \(\mathbb {F}_p\) ${\mathbb{F}}_p$ thus reducing the stable finiteness conjecture to the case of \(\mathbb {F}_p\) ${\mathbb{F}}_p$ . A further application of this result shows that the validity of Kaplansky’s stable finiteness conjecture is equivalent to a version of Gottschalk’s surjunctivity conjecture for additive cellular automata.