International Journal of Algebra and Computation

We examine the construction of Huang and Osajda that was used in their proof of the biautomaticity of Artin groups of almost large type. We describe a slightly simpler variant of that biautomatic structure, with explicit descriptions of a few small examples, and we examine some of the properties of the structure. We explain how the construction can be programmed within the GAP system.

We consider expansive group actions on a compact metric space containing a special fixed point denoted by $0$, and endomorphisms of such systems whose forward trajectories are attracted toward $0$. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to $0$ in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in particular shows that the class includes all residually finite solvable groups and all groups of polynomial growth. For expansive actions of the group $\mathbb{Z}$, we show that a very weak gluing property suffices for nil-rigidity. For ${\mathbb{Z}}^2$-subshifts of finite type, we show that the block-gluing property suffices. The study of nil-rigidity is motivated by two aspects of the theory of cellular automata and symbolic dynamics: It can be seen as a finiteness property for groups, which is representative of the theory of cellular automata on groups. Nilpotency also plays a prominent role in the theory of cellular automata as dynamical systems. As a technical tool of possible independent interest, the proof involves the construction of tiered dynamical systems where several groups act on nested subsets of the original space.

We apply Reeh’s methods of determining fusion-stable finite sets together with Boltje’s integrality conditions to determine stable elements in the $A$-fibered Burnside group of a $p$-group under the action of a saturated fusion system ${ℱ}$.

In this work, we extend an established isomorphism between group rings and a subring of the $n\times n$ matrices. This extension allows us to construct more complex matrices over the ring $R.$ We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism.

In this paper, we use the notion of twisted subgroups (i.e. subsets of group elements closed under the binary operation $(a,b)\mapsto aba$) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus–Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.

In this paper, we study a class of infinitesimal deformations of the centerless higher rank Heisenberg–Virasoro algebras. Explicitly, the universal central extensions, derivations and isomorphism classes of these algebras are determined.

Let $0<\lambda \le 1$, $\lambda \notin \{\frac{2}{4},\frac{2}{7},\frac{2}{10},\frac{2}{13},\dots \}$, be a real and $p$ a prime number, with $[p,p+\lambda p]$ containing at least two primes. Denote by $f_{\lambda }(p)$ the largest integer which cannot be written as a sum of primes from $[p,p+\lambda p]$. Then

Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I(D),I(G)$ be the edge ideals corresponding to $D$ and $G,$ respectively. We give an explicit description of the symbolic powers of $I(D)$ using the concept of strong vertex covers. We show that the ordinary and symbolic powers of $I(D)$ and $I(G)$ behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of $I(D)$ for certain classes of weighted oriented graphs. When $D$ is a weighted oriented odd cycle, we compute $\mathrm{\text{reg}}(I{(D)}^{(s)}/I{(D)}^s)$ and prove $\mathrm{\text{reg}}I{(D)}^{(s)}\le \mathrm{\text{reg}}I{(D)}^s$ and show that equality holds when there is only one vertex with non-trivial weight.

We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most $8$. This result has applications to equations over torsion-free groups: in particular a singular equation of syllable length at most $18$ over a torsion-free group has a solution in some overgroup.