A note on morphisms to wreath products
Year: 2025
Type: article
Abstract: Given a morphism $\varphi \;:\; G \to A \wr B$ from a finitely presented group G to a wreath product $A \wr B$ , we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free subgroup and $\varphi$ factors through an acylindrically hyperbolic quotient of G . As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a finitely presented group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier–Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings.
Authors: ANTHONY GENEVOIS, ROMAIN TESSERA
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4286897088