In this paper, we investigate the set of accumulation points of normalized roots of infinite Coxeter groups for certain class of their action. Concretely, we prove the conjecture proposed in [6, Section 3.2] in the case where the equipped Coxeter matrices are of type $(n-1,1)$, where $n$ is the rank. Moreover, we obtain that the set of such accumulation points coincides with the closure of the orbit of one point of normalized limit roots. In addition, in order to prove our main results, we also investigate some properties on fixed points of the action.

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

Let $k$ be a nonnegative integer. A subgroup $X$ of a group $G$ has normal length $k$ in $G$ if all chains between $X$ and its normal closure $X^{G}$ have length at most $k$, and $k$ is the length of at least one of these chains. The group $G$ is said to have finite normal length if there is a finite upper bound for the normal lengths of its subgroups. The aim of this paper is to study groups of finite normal length. Among other results, it is proved that if all subgroups of a locally (soluble-by-finite) group $G$ have finite normal length in $G$, then the commutator subgroup $G^{\prime }$ is finite and so $G$ has finite normal length. Special attention is given to the structure of groups of normal length $2$. In particular, it is shown that finite groups with this property admit a Sylow tower.

For a group G, a weak Cayley table isomorphism is a bijection f : G → G such that f(g1g2) is conjugate to f(g1)f(g2) for all g1, g2 ∈ G. The set of all weak Cayley table isomorphisms forms a group (G) that is the group of symmetries of the weak Cayley table of G. We determine (G) for each of the 17 wallpaper groups G, and for some other crystallographic groups.

For an element g of a group G, an Engel sink is a subset ${\mathscr E}$(g) such that for every x ∈ G all sufficiently long commutators [. . .[[x, g], g], . . ., g] belong to ${\mathscr E}$(g). A finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group G every element has an Engel sink generating a subgroup of rank r, then G has a normal subgroup N of rank bounded in terms of r such that G/N is nilpotent.

Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$, the exterior current algebra of $\mathfrak{sl}_{2}$. When $\mathbb{L}$ is an $m$-framed $n$-cable of a knot $K\subset S^{3}$, its sutured annular Khovanov homology carries a commuting action of the symmetric group $\mathfrak{S}_{n}$. One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical $\mathfrak{sl}_{2}$ Schur–Weyl duality when $K$ is the Seifert-framed unknot.

We use elementary skein theory to prove a version of a result of Stylianakis (Stylianakis, The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere, arXiv:1511.02912) who showed that under mild restrictions on m and n, the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.

We discuss the internal structure of graph products of right LCM semigroups and prove that there is an abundance of examples without property (AR). Thereby we provide the first examples of right LCM semigroups lacking this seemingly common feature. The results are particularly sharp for right-angled Artin monoids.

Let a prime $p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a $p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the $p$-Sylow subgroups are the groups of type $F_{4}$ when $p=2$ and $I_{2}(m)$ when $m\geq 6$ is even but not a power of $2$ for each odd prime divisor $p$ of $m$. The classification significantly reduces the cases required to describe the $p$-Sylow subgroups of finite real reflection groups.

We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, that is, continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.

We determine the lower rank of the direct product of finitely many hereditarily just infinite profinite groups of finite lower rank.

In this paper, we consider how to express an Iwahori–Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman–Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group; this corrects a result of Bump–Nakasuji.

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.

Let $G$ be a group hyperbolic relative to a finite collection of subgroups ${\mathcal{P}}$. Let ${\mathcal{F}}$ be the family of subgroups consisting of all the conjugates of subgroups in ${\mathcal{P}}$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{{\mathcal{F}}}G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron’s conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups.

Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

We introduce coarse flow spaces for relatively hyperbolic groups and use them to verify a regularity condition for the action of relatively hyperbolic groups on their boundaries. As an application the Farrell–Jones conjecture for relatively hyperbolic groups can be reduced to the peripheral subgroups (up to index-2 overgroups in the $L$-theory case).

Asymptotic triangulations can be viewed as limits of triangulations under the action of the mapping class group. In the case of the annulus, such triangulations have been introduced in K. Baur and G. Dupont (Compactifying exchange graphs: Annuli and tubes, Ann. Comb.3(18) (2014), 797–839). We construct an alternative method of obtaining these asymptotic triangulations using Coxeter transformations. This provides us with an algebraic and combinatorial framework for studying these limits via the associated quivers.