We show that if G is an amenable group and H is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$.

Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.

We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means of invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$. In particular, we improve bounds found by Newman for the order of $M(2,2^k)$. We study identities in $M(2,n)$, which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$.

We study the number of ways of factoring elements in the complex reflection groups$G(r,s,n)$ as products of reflections. We prove a result that compares factorization numbers in$G(r,s,n)$ to those in the symmetric group$S_n$, and we use this comparison, along with the Ekedahl, Lando, Shapiro, and Vainshtein (ELSV) formula, to deduce a polynomial structure for factorizations in$G(r,s,n)$.

Let $m\leqslant n\in \mathbb {N}$, and $G\leqslant \operatorname {Sym}(m)$ and $H\leqslant \operatorname {Sym}(n)$. In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$ and ${V_n(H)}$. When $n\equiv 1 \mod (m-1)$, and under some other technical conditions, we find an embedding of ${V_n(H)}$ into ${V_m(G)}$ via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$ and an embedding of ${V_m(G)}$ into ${V_n(H)}$.

Given an integer $g>2$, we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

For a Coxeter system and a representation $V$ of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when $V$ is reflection faithful. Elias and Williamson defined another category when $V$ is not reflection faithful and proved that this category is equivalent to the category of Soergel bimodules when $V$ is reflection faithful. Moreover, they proved the categorification theorem for their category with fewer assumptions on $V$. In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$.

We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.

We improve a recent construction of Andrés Navas to produce the first examples of $C^2$-undistorted diffeomorphisms of the interval that are $C^{1+\alpha }$-distorted (for every ${\alpha < 1}$). We do this via explicit computations due to the failure of an extension to class $C^{1+\alpha }$ of a classical lemma related to the work of Nancy Kopell.

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$. However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers.

Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author.

A subgroup H of a group G is said to be contranormal in G if the normal closure of H in G is equal to G. In this paper, we consider groups whose nonmodular subgroups (of infinite rank) are contranormal.

L'analyse de la démonstration par contradiction de Frécon 2018 qui est faite dans Poizat 2018 met en évidence la structure symétrique des groupes de rang de Morley fini sans involutions; en effet, cette démonstration consiste en la construction d'un espace symétrique de dimension deux (“un plan”), puis à montrer que ce plan ne peut exister.

Aux sous-espaces symétriques définissables de ces groupes sont associées des symétries et des transvexions, qu'on entreprend d'étudier ici dans l'abstrait, sans référence à un groupe qui les enveloppe; cela nous mène à considérer des structures introduites axiomatiquement que nous appelons symétrons (plutôt qu'ensembles symétriques diadiques, comme les ont nommées Lawson & Lim 2004).

Le $Z^*$-Theorem de Glauberman permet d'élucider complètement la structure des symétrons finis: chacun est isomorphe à l'ensemble des symétries associées à un sous-espace symétrique d'un groupe fini sans involutions, qui est loin d'être uniquement déterminé: de fait, il existe des groupes finis non isomorphes qui ont les mêmes symétries, et aussi des symétrons finis qui ne sont pas isomorphes aux symétries d'un groupe,

La situation est plus incertaine dans le cas des symétrons de rang de Morley fini, ou même algébriques, qui sont l'objet d'étude principal de cet article. Mais bien qu'un symétron soit une structure nettement plus faible qu'un groupe, nous pouvons étendre aux symétrons des résultats bien connus à propos des groupes de rang de Morley fini: condition de chaîne, décomposition en composantes connexes, caractérisation des parties définissables génériques, génération elliptique, etc. Ces propriétés sont nouvelles même dans le cas des sous-espaces symétriques d’un groupe, et permettent de court-circuiter les calculs de Frécon dans la construction de son plan paradoxal.

En outre, sous l'hypothèse de la Conjecture d'Algébricité, nous généralisons le Théorème de Glauberman au contexte de rang de Morley fini.

The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. We consider this type of systems for the case of ordered triples and express the size of the solution set in terms of the irreducible characters of the group. The obtained formulas are natural extensions of Frobenius’ character formula that calculates the number of ways a group element is a commutator of an ordered pair of elements in a finite group. We discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and we show explicit calculations for dihedral groups.

We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces.

More precisely, let $\Sigma $ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that ${\mathrm {Map}}(\Sigma )$ admits a continuous nonelementary action on a hyperbolic space if and only if $\Sigma $ contains a finite-type subsurface which intersects all its homeomorphic translates.

When $\Sigma $ contains such a nondisplaceable subsurface K of finite type, the hyperbolic space we build is constructed from the curve graphs of K and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of ${\mathrm {Map}}(\Sigma )$ contains an embedded $\ell ^1$; second, using work of Dahmani, Guirardel and Osin, we deduce that ${\mathrm {Map}} (\Sigma )$ contains nontrivial normal free subgroups (while it does not if $\Sigma $ has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal.

Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$. More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$.