International Journal of Algebra and Computation

For any finite group A and any finitely generated group B, we prove that the corresponding lamplighter group $A\wr B$ admits a standard generating set with unbounded depth, and that if B is abelian then the above is true for every standard generating set. This generalizes the case where $B=\mathbb{Z}$ together with its cyclic generator [S. Cleary and J. Taback, Dead end words in lamplighter groups and other wreath products, Q. J. Math. 56(2) (2005) 165–178]. When $B=H\ast K$ is the free product of two finite groups H and K, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.

We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a nontrivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras $33_{37},35_{37}$, $77_{83}$, $78_{83}$, $80_{83}$, $82_{83}$, $83_{83}$, $1310_{1316}$, $1313_{1316}$, $1315_{1316}$ and $1316_{1316}$. Prior to our paper, only $83_{83}$ and $1316_{1316}$ were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed anti-Ramsey scheme. The notion of a directed anti-Ramsey scheme may be of independent interest.

We complement our upper bounds with some lower bounds. Namely, we show that any square representation of $31_{37}$ requires at least 14 points, any square representation of $33_{37}$ requires at least 11 points, and any square representation of $35_{37}$ requires at least 12 points. Our technique adapts previous work of Alm et al. [Algebra Univ. (2022)], in that we examine the combinatorial structure induced by the flexible atom.

In this paper, we explore some applications of a certain technique (that we call Freese’ technique), which is a tool for identifying certain lattices as sublattices of the congruence lattice of a given algebra. In particular we will give sufficient conditions for two family of lattices (called the rods and the snakes) to be admissible as sublattices of a variety generated by a given algebra, extending an unpublished result of Freese and Lipparini.

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: $F_n{⋊}_{\phi }\mathbb{Z}$. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for $n=2$ the length of the lower central series of $F_n{⋊}_{\phi }\mathbb{Z}$ is equal to 2, $\omega$ or ${\omega }^2$.

An important theorem by Baker and Pixley states that if $A$ is a finite algebra with a $(d+1)$-ary near-unanimity term and $f:A^n\to A$ is any function, then f is a term-function of $A$ if f preserves all subuniverses of $A^d$. This result was generalized recently for (classes of) not necessarily finite algebras using sheaf-theoretic tools. In this work, we give a short model-theoretic proof of this generalization. Also, we apply the theorem to obtain a characterization of dual discriminator varieties, and to give necessary and sufficient conditions for a variety with a near-unanimity term to be congruence permutable.

Principal fully irreducible outer automorphisms were introduced in [Y. Algom-Kfir, I. Kapovich and C. Pfaff, Stable strata of geodesics in outer space, Int. Math. Res. Not. 2019(14) (2019) 4549–4578] to emulate principal pseudo-Anosov surface homeomorphisms, i.e. those whose attracting and repelling invariant foliations have only 3-pronged singularities. It is proved in [Y. Algom-Kfir, I. Kapovich and C. Pfaff, Stable strata of geodesics in outer space, Int. Math. Res. Not. 2019(14) (2019) 4549–4578] that these outer automorphisms semi-mimic principal pseudo-Anosov surface homeomorphisms in important ways. We prove here which rank-3 graphs carry train track representatives of principal fully irreducible outer automorphisms. As a corollary, one obtains which simplices of rank-3 Culler–Vogtmann outer space principal axes pass through.

We compare several natural notions of effective presentability of a topological space up to homeomorphism. We note that every left-c.e. (lower-semicomputable) Stone space is homeomorphic to a computable one. In contrast, we produce an example of a locally compact, left-c.e. space that is not homeomorphic to any computable Polish space. We apply a similar technique to produce examples of computable topological spaces not homeomorphic to any right-c.e. (upper-semicomputable) Polish space, and indeed to any arithmetical or even analytical Polish space. We then apply our techniques to totally disconnected locally compact (tdlc) groups. We prove that every computably locally compact tdlc group is topologically isomorphic to a computable tdlc group.

Let E be the infinite-dimensional Grassmann algebra over an infinite field F of characteristic different from 2. The main purpose of this paper is to describe the $T_{\mathbb{Z}}$-ideal of the graded polynomial identities and the $T_{\mathbb{Z}}$-space of the central polynomials of E equipped with its $2$ and 3-induced $\mathbb{Z}$-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, $\mathbb{Z}$-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on $\mathbb{Z}$-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.