We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional (4D) topological surgery conjecture. The A–B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surgery problems, defined using ramifications of homotopically trivial links. More generally we show how $n$ -Engel relations arise from higher-order double points of surfaces in 4-space.

A compact set K ⊂ ℝn is Whitney 1-regular if the geodesic distance in K is equivalent to the Euclidean distance. Let P be the Chevalley map defined by an integrity basis of the algebra of polynomials invariant by a reflection group. This paper gives the Whitney 1-regularity of the image by P of any closed ball centred at the origin. The proof uses the works of Givental', Kostov and Arnol'd on the symmetric group. It needs a generalization of a property of the Vandermonde determinants to the Jacobian of the Chevalley mappings.

This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

Suppose that a finite group G admits an automorphism of order 2n such that the fixed-point subgroup of the involution is nilpotent of class c. Let m = ) be the number of fixed points of . It is proved that G has a characteristic soluble subgroup of derived length bounded in terms of n, c whose index is bounded in terms of m, n, c. A similar result is also proved for Lie rings.

In this paper, we study Abelian and metabelian quotients of braid groups of oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell–Neuwirth fibrations.

We study representations of the loop braid group LB n from the perspective of extending representations of the braid group $\mathcal{B}$ n . We also pursue a generalization of the braid/Hecke/Temperlely–Lieb paradigm – uniform finite dimensional quotient algebras of the loop braid group algebras.

We give a means of estimating the equivariant compression of a group G in terms of properties of open subgroups G i ⊂ G whose direct limit is G. Quantifying a result by Gal, we also study the behaviour of the equivariant compression under amalgamated free products G 1∗H G 2 where H is of finite index in both G 1 and G 2.

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$ -group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$ -group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$ .

It is shown that for many branch groups $G$ the action on the ambient tree can be interpreted in $G$ , in the sense of first-order model theory.

We prove that either the images of the mappingclass groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further, we show that the images of the mapping class groups have non-trivial 2-cohomology, at least for small levels. For this purpose, we considered a series of quasi-homomorphisms on mapping class groups extending the previous work of Barge and Ghys (Math. Ann. 294 (1992), 235–265) and of Gambaudo and Ghys (Bull. Soc. Math. France 133(4) (2005), 541–579). These quasi-homomorphisms are pull-backs of the Dupont–Guichardet–Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations.

We give an alternate proof of Wise’s malnormal special quotient theorem (MSQT), avoiding cubical small cancelation theory. We also show how to deduce Wise’s Quasiconvex Hierarchy Theorem from the MSQT and theorems of Hsu and Wise and Haglund and Wise.

Let m, n be positive integers, let υ be a multilinear commutator word and let w = υ m . We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).

Supplementary materials are available with this article.

We present a careful approximation of the quasi-geodesics of trees of hyperbolic and relatively hyperbolic spaces. As an application we prove a dynamical and geometric combination theorem for trees of relatively hyperbolic spaces, with both Farb's and Gromov's definitions.

Groups of infinite rank in which every subgroup is either normal or contranormal are characterised in terms of their subgroups of infinite rank.

We extend to soluble $\text{FC}^{\ast }$-groups, the class of generalised FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J. 28(3) (2002), 241–254], the characterisation of finite soluble T-groups obtained recently in Kaplan [‘On T-groups, supersolvable groups, and maximal subgroups’, Arch. Math. (Basel) 96(1) (2011), 19–25].

This note completes the proof of the structure theorem for pp-matroid groups which was stated in our earlier paper J. Krempa and A. Stocka [‘On sets of pp-generators of finite groups’, Bull. Aust. Math. Soc.91 (2015), 241–249].

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank-$n$ group $G$ and $H$ has index at least $n$ in $G$, we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$; this construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n\leq 3$, there is a simultaneous left–right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank-$n$ group $G$ with index less than $3\cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H\cong C_{2}^{n}$.

In this paper, we prove that the finite simple groups $\text{PSp}_{6}(q)$, ${\rm\Omega}_{7}(q)$ and $\text{PSU}_{7}(q^{2})$ are $(2,3)$-generated for all $q$. In particular, this result completes the classification of the $(2,3)$-generated finite classical simple groups up to dimension 7.

We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.