Hopfian wreath products and the stable finiteness conjecture (Q6624132)

Let \(\Delta\) and \(\Gamma\) be two discrete groups. Their (restricted) wreath product is the group \(\Delta \wr \Gamma= \big (\bigoplus_{\Gamma} \Delta \big) \rtimes \Gamma \), where \(\Gamma\) acts on the direct sum by shifting coordinates. One of the first and influential occurrences of wreath products in combinatorial group theory was in a study by \textit{K. W. Gruenberg} [Proc. Lond. Math. Soc., III. Ser. 7, 29--62 (1957; Zbl 0077.02901)] of residual properties of solvable groups. Along the way, he showed that a wreath product of two groups \(\Delta \wr \Gamma\) is residually finite if and only if \(\Delta\) and \(\Gamma\) are residually finite and either \(\Delta\) is abelian or \(\Gamma\) is finite. When \(\Delta\) and \(\Gamma\) are both finitely generated, \(\Delta \wr \Gamma\) is also finitely generated and so in particular if \(\Delta\) is abelian and \(\Gamma\) is residually finite, then \(\Delta \wr \Gamma\) is Hopfian.\N\NIn the paper under review, the authors study the Hopf property for wreath products of finitely generated groups, focusing on the case of an abelian base group. The main result is Theorem 1.3: The following are equivalent: (1) For every finitely generated abelian group \(A\) and every finitely generated Hopfian group \(\Gamma\), the wreath product \(A \wr \Gamma\) is Hopfian. (2) Kaplansky's direct finiteness conjecture holds.\N\NA ring with identity \(R\) is directly finite if every element with a one-sided inverse is a unit; equivalently, if \(xy=1\) implies \(yx=1\). It is stably finite if the matrix rings \(\mathbb{M}_{d}(R)\) are directly finite for all \(d \geq 1\). Kaplansky's direct (resp. stable) finiteness conjecture asserts that the group ring \(\mathbb{F}[\Gamma]\) is directly (resp. stably) finite for every group \(\Gamma\) and every field \(\mathbb{F}\).\N\NOn the opposite end of the spectrum, the authors examine the case in which \(\Delta\) has certain properties that are incompatible with properties of \(\Gamma\). Theorem 1.13: There exist finitely generated groups \(\Delta\), \(\Gamma\) such that \(\Gamma\) is non-Hopfian but \(\Delta \wr \Gamma\) is Hopfian.