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A note on morphisms to wreath products

Anthony Genevois, Romain Tessera

TL;DR

This paper establishes structural constraints on morphisms from finitely presented groups $G$ to wreath products $A\wr B$ by combining truncated wreath presentations with the geometry of graph products. The main theorem shows that such morphisms factor through a quotient of $G$ into either a small (finite-by-$A$) image or a large quotient that is acylindrically hyperbolic (hence SQ-universal), or acts nontrivially on a finite-dimensional CAT(0) cube complex with hyperplane-stabilizers virtually embedded in finite powers of $A$; as a corollary, finitely presented subgroups of $A\wr B$ are classified as either subgroups of $B$ or of $(A^n)\rtimes F$, and any group surjecting onto $A\wr B$ is SQ-universal. The automorphism structure of wreath products is analyzed, yielding an explicit decomposition of $\mathrm{Aut}(F\wr H)$ in favorable cases and connecting automorphisms to units in group rings, with consequences contingent on Kaplansky-type unit conjectures. Collectively, the results illuminate how wreath-product targets constrain domain groups, provide a bridge to negative-curvature techniques, and expose deep links with group-ring units and CAT(0) geometry.

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.

A note on morphisms to wreath products

TL;DR

This paper establishes structural constraints on morphisms from finitely presented groups to wreath products by combining truncated wreath presentations with the geometry of graph products. The main theorem shows that such morphisms factor through a quotient of into either a small (finite-by-) image or a large quotient that is acylindrically hyperbolic (hence SQ-universal), or acts nontrivially on a finite-dimensional CAT(0) cube complex with hyperplane-stabilizers virtually embedded in finite powers of ; as a corollary, finitely presented subgroups of are classified as either subgroups of or of , and any group surjecting onto is SQ-universal. The automorphism structure of wreath products is analyzed, yielding an explicit decomposition of in favorable cases and connecting automorphisms to units in group rings, with consequences contingent on Kaplansky-type unit conjectures. Collectively, the results illuminate how wreath-product targets constrain domain groups, provide a bridge to negative-curvature techniques, and expose deep links with group-ring units and CAT(0) geometry.

Abstract

Given a morphism from a finitely presented group to a wreath product , we show that, if the image of is a sufficiently large subgroup, then contains a non-abelian free subgroup and factors through an acylindrically hyperbolic quotient of . As direct applications, we classify the finitely presented subgroups in up to isomorphism and we deduce that a group having a wreath product 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.

Paper Structure

This paper contains 17 sections, 21 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Nearly injective morphisms.
  3. Nearly surjective morphisms.
  4. Fixed-point property.
  5. Automorphisms.
  6. Acknowledgments.
  7. A few words about graph products of groups
  8. Normal form.
  9. Quasi-median geometry.
  10. Negative curvature.
  11. Proof of the main theorem
  12. Automorphisms
  13. A word about approximating infinitely presented groups
  14. Algebraic approximation.
  15. Geometric approximation.
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $A,B,G$ be three groups. Assume that $G$ is finitely presented. Every morphism $\varphi : G \to A \wr B$ factors through a quotient $\overline{G}$ of $G$ such that: In particular, in the second item, $\overline{G}$ semisplits over a subgroup that virtually embeds in a finite power of $A$; so, if $A$ is finite, then $\overline{G}$ is multi-ended.

Figures (2)

  • Figure 1: Example of a graph $\mathrm{QM}(\Gamma, \mathcal{G})$.
  • Figure 2: Four hyperplanes in a quasi-median graph, coloured in red, blue, green, and orange. The orange hyperplane is transverse to the blue and red hyperplanes.

Theorems & Definitions (44)

  • Theorem 1.1
  • Corollary 1.2
  • proof : Proof of Corollary \ref{['cor:FPSub']}.
  • Corollary 1.3
  • proof
  • Theorem 1.4: MR2764930
  • Example 1.5
  • Proposition 1.6
  • proof : Proof of Proposition \ref{['prop:Burnside']}.
  • Corollary 1.7
  • ...and 34 more