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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 generalise this observation to rotation groups, which can be thought of as a common generalisation 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)., Comment: 26 pages, 4 figures. Comments are welcome!

In this article, we introduce and study a natural notion of coarse separation for metric spaces, with an emphasis on coarse separation by subspaces of polynomial or subexponential growth. For instance, we show that symmetric spaces of non-compact type different from $\mathbb{H}_\mathbb{R}^2$ and thick Euclidean buildings of rank $\geq 2$ cannot be coarsely separated by subspaces of subexponential growth; and that a connected nilpotent Lie group of growth degree $D \geq 2$ cannot be coarsely separated by a subspace of polynomial degree $\leq D-2$. We apply these results to the large-scale geometry of amalgamated free products and wreath products. The latter application is based on an Embedding Theorem that generalises previous work of the last two authors, and which is of independent interest. We also discuss some further applications to the distorsion of coarse embeddings between certain metric spaces., Comment: 53 pages, 8 figures. Comments are welcome!

In this article, we introduce halo products as a natural generalisation of wreath products. They also encompass lampshuffler groups $\mathrm{FSym}(H) \rtimes H$ and lampcloner groups $\mathrm{FGL}(H) \rtimes H$, as well as many possible variations based for instance on braid groups, Thompson's groups, mapping class groups, automorphisms of free groups. We build a geometric framework that allows us to study the large-scale geometry of halo groups, providing refined invariants distinguishing various halo groups up to quasi-isometry., Comment: 83 pages, 8 figures. Comments are welcome!

Recently, Lin and Shinder constructed non-trivial homomorphisms from Cremona groups of rank >2 to \mathbb{Z} using motivic techniques. In this short note we propose an alternative perspective from median geometry on their theorem., Comment: 6 pages, 3 figures

For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct groups satisfying $(\mathrm{FW}_n)$ but not $(\mathrm{FW}_{n+1})$. Our examples are virtually free abelian and virtually graph products of finite groups. As an application of our constructions, we obtain, for every $n \geq 1$, a group satisfying $(\mathrm{FW}_n)$ but acting properly and cocompactly on a median graph of cubical dimension $n+1$. Several conjectures and open questions are included., Comment: 19 pages. Comments are welcome!

Extending Stallings' foldings of trees, we show in this article that every parallel-preserving map between median graphs factors as an isometric embedding through a sequence of elementary transformations which we call foldings and swellings. This new construction proposes a unified point of view on Beeker and Lazarovich's work on folding pocsets and on Ben-Zvi, Kropholler, and Lyman's work on folding nonpositively curved cube complexes., Comment: 44 pages, 5 figures. Comments are welcome!

In this note, we discuss and motivate the use of the terminology ``median graphs'' in place of ``CAT(0) cube complexes'' in geometric group theory., Comment: 10 pages. Comments are welcome!

In this article, we show that, for every $n \geq 2$, the pure virtual twin group $PVT_n$ can be naturally described as a symmetric diagram group, a family of groups introduced by V. Guba and M. Sapir and associated to semigroup presentations. Inspired by this observation, we prove that every finitely generated right-angled Artin group is a symmetric diagram group. This contrasts with the fact that not all right-angled Artin groups are planar diagram groups., Comment: 17 pages, 10 figures. Comments are welcome!

Given a graph $\Gamma$, the right-angled Artin group $A(\Gamma)$ is given by the presentation $\langle u \in V(\Gamma) \mid [u,v]=1, \ \{u,v\} \in E(\Gamma) \rangle$. The Embedding Problem in right-angled Artin groups asks, given two finite graphs $\Phi,\Psi$, how to determine whether or not $A(\Phi)$ is isomorphic to a subgroup of $A(\Psi)$. These are the notes of a mini-course given at the CIRM in April 2022, dedicated to a geometric point of view on this embedding problem, based on quasi-median graphs., Comment: 20 pages, 11 figures. Notes of a mini-course given at the CIRM in April 2022 for the thematic semester "Groups Acting on Fractals, Hyperbolicity and Self-Similarity" of the Institut Henri Poincar\'e

We introduce Property (NL), which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. In other words, such a group $G$ can only admit elliptic or horocyclic hyperbolic actions, and consequently its poset of hyperbolic structures $\mathcal{H}(G)$ is trivial. It turns out that many groups satisfy this property; and we initiate the formal study of this phenomenon. Of particular importance is the proof of a dynamical criterion in this paper that ensures that groups with rich actions on compact Hausdorff spaces have Property (NL). These include many Thompson-like groups, such as $V, T$ and even twisted Brin--Thompson groups, which implies that every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property (NL). We also study the stability of the property under group operations and explore connections to other fixed point properties. In the appendix (by Alessandro Sisto) we describe a construction of cobounded actions on hyperbolic spaces starting from non-cobounded ones that preserves various properties of the initial action., Comment: 48 pages. Main paper by the first three authors, appendix by the fourth author. v3: final version, to appear in Groups, Geometry and Dynamics

In this article, we introduce rotation groups as a common generalisation of Coxeter groups and graph products of groups (including right-angled Artin groups). We characterise algebraically these groups by presentations (periagroups) and we propose a combinatorial geometry (mediangle graphs) to study them. As an application, we give natural and unified proofs for several results that hold for both Coxeter groups and graph products of groups., Comment: 39 pages, comments are welcome!

Cactus groups and their pure subgroups appear in various fields of mathematics and are currently attracting attention from diverse mathematical communities. They share similarities with both right-angled Coxeter groups and braid groups. In this article, our goal is to highlight the tools offered by geometric group theory for the study of these groups. Among the new contributions made possible thanks to this geometric perspective, we describe an explicit and efficient solution to the conjugacy problem, and we prove that cactus groups are virtually cocompact special and acylindrically hyperbolic., Comment: 33 pages, comments are welcome!

To every semigroup presentation $\mathcal{P}= \langle \Sigma \mid \mathcal{R} \rangle$ and every baseword $w \in \Sigma^+$ can be associated a diagram group $D(\mathcal{P},w)$, defined as the fundamental group of the so-called Squier complex $S(\mathcal{P},w)$. Roughly speaking, $D(\mathcal{P},w)$ encodes the lack of asphericity of $\mathcal{P}$. Examples of diagram groups include Thompson's group $F$, the lamplighter group $\mathbb{Z} \wr \mathbb{Z}$, the pure planar braid groups, and various right-angled Artin groups. This survey aims at summarising what is known about the family of diagram groups., Comment: 67 pages, comments are welcome!

Given a quasi-median graph $X$, the crossing graph $\Delta X$ and the contact graph $\Gamma X$ are natural hyperbolic models of $X$. In this article, we show that the asymptotic translation length in $\Delta X$ or $\Gamma X$ of an isometry of $X$ is always rational. Moreover, if $X$ is hyperbolic, these rational numbers can be written with denominators bounded above uniformly; this is not true in full generality. Finally, we show that, if the quasi-median graph $X$ is constructible in some sense, then there exists an algorithm computing the translation length of every computable isometry. Our results encompass contact graphs in CAT(0) cube complexes and extension graphs of right-angled Artin groups., Comment: 35 pages, 6 figures. Comments are welcome

We exhibit a variety of groups that act properly and even cocompactly on median graphs (a.k.a. one-skeletons of CAT(0) cube complexes), with quasi-isometric groups that do not admit any proper action on a median graph. This answes a question of Niblo, Sageev and Wise. Our examples are all quasi-isometrically trivial central extensions of certain cubulated groups., Comment: 13 pages. v2: introduction slightly rewritten, with added references. v3: final version. To appear in Proceedings of the AMS

We show that plane Cremona groups over finite fields embed as dense subgroups into Neretin groups, i.e. groups of almost automorphisms of rooted trees. We also show that if the finite base field has even characteristic and contains at least 4 elements, then the permutations induced by birational transformations on rational points of regular projective surfaces are even. In a second part, we construct explicit locally compact CAT(0) cube complexes, on which Neretin groups act properly. This allows us to recover in a unified way various results on Neretin groups such as that they are of type $F_{\infty}$. We also prove a new fixed-point theorem for CAT(0) cube complexes without infinite cubes and use it to deduce a regularisation theorem for plane Cremona groups over finite fields., Comment: 30 pages, 1 figure. Updated version taking into consideration the comments of the referee, minor issued fixed. Accepted in IMRN

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., Comment: 20 pages, 2 figures. Comments are welcome!

In this article, we introduce a new family of groups, called Chambord groups and constructed from braided strand diagrams associated to specific semigroup presentations. It includes the asymptotically rigid mapping class groups previously studied by the authors such as the braided Higman-Thompson groups and the braided Houghton groups. Our main result shows that polycyclic subgroups in Chambord groups are virtually abelian and undistorted., Comment: 47 pages, 23 figures. Minor revisions. Accepted for publication in Transactions of the AMS

It is known that a cocompact special group $G$ does not contain $\mathbb{Z} \times \mathbb{Z}$ if and only if it is hyperbolic; and it does not contain $\mathbb{F}_2 \times \mathbb{Z}$ if and only if it is toric relatively hyperbolic. Pursuing in this direction, we show that $G$ does not contain $\mathbb{F}_2 \times \mathbb{F}_2$ if and only if it is weakly hyperbolic relative to cyclic subgroups, or cyclically hyperbolic for short. This observation motivates the study of cyclically hyperbolic groups, which we initiate in the class of groups acting geometrically on CAT(0) cube complexes. Given such a group $G$, we first prove a structure theorem: $G$ virtually splits as the direct sum of a free abelian group and an acylindrically hyperbolic cubulable group. Next, we prove a strong Tits alternative: every subgroup $H \leq G$ either is virtually abelian or it admits a series $H=H_0 \rhd H_1 \rhd \cdots \rhd H_k$ where $H_k$ is acylindrically hyperbolic and where $H_i/H_{i+1}$ is finite or free abelian. As a consequence, $G$ is SQ-universal and it cannot contain subgroups such that products of free groups and virtually simple groups., Comment: 34 pages, 7 figures. Comments are welcome