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 ${{\mathbb R}}$ .

We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

Let $M(A,n)$ be the Moore space of type $(A,n)$ for an Abelian group A and $n\ge 2$ . We show that the loop space $\Omega (M(A,n))$ is homotopy nilpotent if and only if A is a subgroup of the additive group $\mathbb {Q}$ of the field of rationals. Homotopy nilpotency of loop spaces $\Omega (M(A,1))$ is discussed as well.

In this paper, we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm {IA}_3$ of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of $\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm {IA}_3$ has finite index in that of the Andreadakis–Johnson filtration of $\mathrm {IA}_3$.

We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$-algebras arising from minimal free $\mathbb {Z}$-actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

We construct a sofic approximation of ${\mathbb F}_2\times {\mathbb F}_2$ that is not essentially a ‘branched cover’ of a sofic approximation by homomorphisms. This answers a question of L. Bowen.

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Given a commutative unital ring R, we show that the finiteness length of a group G is bounded above by the finiteness length of the Borel subgroup of rank one $\textbf {B}_2^{\circ }(R)=\left ( \begin {smallmatrix} * & * \\ 0 & * \end {smallmatrix}\right )\leq \operatorname {\textrm {SL}}_2(R)$ whenever G admits certain R-representations with metabelian image. Combined with results due to Bestvina–Eskin–Wortman and Gandini, this gives a new proof of (a generalization of) Bux’s equality on the finiteness length of S-arithmetic Borel groups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels’ groups $\textbf {A}_n(R) \leq \operatorname {\textrm {GL}}_n(R)$ in terms of n and $\textbf {B}_2^{\circ }(R)$ . This generalizes earlier results due to Remeslennikov, Holz, Lyul’ko, Cornulier–Tessera, and points out to a conjecture about the finiteness length of such groups.

We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

We consider the Birman–Hilden inclusion $\phi\colon\Br_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\phi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\phi^*(H_1(\Sigma_{g,1}))$ has only 4-torsion.

Let W be a 2-dimensional Coxeter group, that is, one with 1/mst + 1/msr + 1/mtr ≤ 1 for all triples of distinct s, t, r ∈ S. We prove that W is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary W), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Świątkowski, groups acting properly and cocompactly on buildings of type W are also biautomatic. We also show that the fellow traveller property for the natural language fails for $W=\widetilde {A}_3$.

This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.

The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

We show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call random Gromov’s monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov’s monsters arising from graphical small cancellation labellings of expanders.

Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov’s monsters.

Let G be a finite group and $\psi (G) = \sum _{g \in G} o(g)$ , where $o(g)$ denotes the order of $g \in G$ . There are many results on the influence of this function on the structure of a finite group G.

In this paper, as the main result, we answer a conjecture of Tărnăuceanu. In fact, we prove that if G is a group of order n and $\psi (G)>31\psi (C_n)/77$ , where $C_n$ is the cyclic group of order n, then G is supersolvable. Also, we prove that if G is not a supersolvable group of order n and $\psi (G) = 31\psi (C_n)/77$ , then $G\cong A_4 \times C_m$ , where $(m, 6)=1$ .

Finally, Herzog et al. in (2018, J. Algebra, 511, 215–226) posed the following conjecture: If $H\leq G$ , then $\psi (G) \unicode[stix]{x02A7D} \psi (H) |G:H|^2$ . By an example, we show that this conjecture is not satisfied in general.

We say that a group G of local (maybe formal) biholomorphisms satisfies the uniform intersection property if the intersection multiplicity $(\phi (V), W)$ takes only finitely many values as a function of G for any choice of analytic sets V and W of complementary dimension. In dimension $2$ we show that G satisfies the uniform intersection property if and only if it is finitely determined – that is, if there exists a natural number k such that different elements of G have different Taylor expansions of degree k at the origin. We also prove that G is finitely determined if and only if the action of G on the space of germs of analytic curves has discrete orbits.

The pronorm of a group G is the set $P(G)$ of all elements $g\in G$ such that X and $X^g$ are conjugate in ${\langle {X,X^g}\rangle }$ for every subgroup X of G. In general the pronorm is not a subgroup, but we give evidence of some classes of groups in which this property holds. We also investigate the structure of a generalised soluble group G whose pronorm contains a subgroup of finite index.