A note on morphisms to wreath products

TL;DR

This paper investigates the structure of morphisms from finitely presented groups to wreath products, revealing conditions under which kernels contain free subgroups and how such morphisms factor through hyperbolic quotients, with applications to subgroup classification and automorphism groups.

Contribution

It establishes new criteria for morphisms to wreath products, classifies finitely presented subgroups, and links automorphism group structures to the Kaplansky conjecture.

Findings

Kernels contain non-abelian free subgroups under certain conditions

Finitely presented subgroups in wreath products are classified up to isomorphism

Automorphism groups of wreath products are characterized, connecting to the Kaplansky conjecture

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 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.