In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $${{\,\mathrm{F}\,}}_\infty $$ . The proof utilized certain contractible cube complexes, which in this paper prove $${{\,\mathrm{CAT}\,}}(0)$$ We then use fact to compute geometric invariants $$\Sigma ^m(F_{{\text {br}}})$$ pure group $$F_{{\text {br}}}$$ Only first invariant ^1...

We introduce a new notion of commutator which depends on a choice of subvariety in any variety of Ω-groups. We prove that this notion encompasses Higgins’s commutator, Fröhlich’s central extensions and the Peiffer commutator of precrossed modules.

A group word w is said to be strongly concise in a class C $\mathcal C$ of profinite groups if, for any G , either takes at least continuum many values or the verbal subgroup ( ) $w(G)$ finite. It conjectured that all words are groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture multilinear commutator words, as well some other particular words. They also every nilpotent group...

We investigate commutator operations on compatible uniformities. We present a commutator operation for uniformities in the congruence-modular case which extends the commutator on congruences, and explore its properties. Introduction The purpose of this paper is to generalize the commutator of congruences to a commutator of compatible uniformities. Commutator theory (on congruences) works best f...

In this article we construct an irreducible simple subgroup G = 〈q, y, t, w〉 of GL8671(13) from an irreducible subgroup T of GL11(2) isomorphic to Mathieu’s simple group M24 by means of Algorithm 2.5 of [13]. We also use the first author’s similar construction of Fischer’s sporadic simple group G1 = Fi23 described in [11]. He starts from an irreducible subgroup T1 of GL11(2) contained in T whic...

В работе построено разрешенное относительно неизвестных уравнение в свободной группе ранга 2, левая часть которого принадлежит коммутанту (имеет нулевую сумму показателей по каждой переменной), а правая - простейший коммутатор, не имеющее решения этой группе, но решение любом конечном гомоморфном образе группы. We construct an equation (resolved with respect to unknowns) such that (i) it has no...

Given a binary quasigroup $G$ of order $n$, $d$-iterated $G[d]$ is the $(d+1)$-ary equal to $d$-times composition with itself. The Cayley table every $d$-ary $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as coincide those corresponding hypercube.
 We prove that if group $n$ satisfies Hall–Paige condition, then number transversals $ \frac...

We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. These theories are used to show that each of the following sets of statements are equivalent for a variety V of algebras. (I) (a) V satisfies a nontrivial congruence identity. (b) V satisfies an idempotent Maltsev cond...

Let Λ $\Lambda$ be a row-finite and source-free higher rank graph with finitely many vertices. In this paper, we define the Higman–Thompson-like group h t $\operatorname{\Lambda _{ht}}$ of C*-algebra O $\mathcal {O}_\Lambda$ to special subgroup unitary in . It is shown that closely related topological full groups groupoid associated Some properties are also investigated. We show its commutator ...