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.

It is a theorem due to F. Haglund and D. Wise that reflection groups (aka Coxeter groups) virtually embed into right-angled reflection groups (aka right-angled Coxeter groups). In this article, we generalize this observation to rotation groups, which can be thought of as a common generalization of Coxeter groups and graph products of groups. More precisely, we prove that rotation groups (aka periagroups) virtually embed into right-angled rotation groups (aka graph products of groups).

From any two median spaces $X$ and $Y$, we construct a new median space $X \circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show that this construction is compatible with wreath products in the following sense: given two finitely generated groups $G,\,H$ and two (equivariant) coarse embeddings into median spaces $X,\,Y$, there exist a(n equivariant) coarse embedding $G\wr H \to X \circledast Y$. The construction offers a unified point of view on various questions related to the Hilbertian geometry of wreath products of groups.

In this article, we generalize Haglund and Wise’s theory of special cube complexes to groups acting on quasi-median graphs. More precisely, we define special actions on quasi-median graphs, and we show that a group which acts specially on a quasi-median graph with finitely many orbits of vertices must embed as a virtual retract into a graph product of finite extensions of clique-stabilizers. In the second part of the article, we apply the theory to fundamental groups of some graphs of groups called right-angled graphs of groups.

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.