Hopfian wreath products and the stable finiteness conjecture

2022/11/02 by Bradford, Henry, Fournier-Facio, Francesco · 1 citation
#FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2211.01510

Abstract

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 Γ the wreath product A \wr Γ is Hopfian. In fact, we characterize precisely when A \wr Γ is Hopfian, in terms of the existence of one-sided units in certain matrix algebras over \mathbbF_p[Γ], 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 \mathbbF_p thus reducing the stable finiteness conjecture to the case of \mathbbF_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.

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