International Journal of Algebra and Computation

We describe subgroups and overgroups of the generalized Thompson groups $V_n$ which arise via conjugation by rational homeomorphisms of Cantor space. We specifically consider conjugating $V_n$ by homeomorphisms induced by synchronizing transducers and their inverses. Our descriptions of the subgroups and overgroups use properties of the conjugating transducer to either restrict or augment the action of $V_n$ on Cantor space. We also consider a class $𝔓$ of transducers containing all invertible, initial transducers. We associate to every transducer $T$ in this class, a group ${\mathscr{H}}(T)$. We show that for a certain subclass of $𝔓$, the groups ${\mathscr{H}}(T)$ are simple.

Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $\mathrm{\text{gr}}(H,A,B)$ be the growth function of the double cosets $AhB,h\in H$. We prove that the behavior of $\mathrm{\text{gr}}(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $\mathrm{\text{gr}}(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups $A$ and $B$ of infinite index, $\mathrm{\text{gr}}(H,A,B)$ is exponential. Moreover, there exists a constant $\lambda >0$, such that $\mathrm{\text{gr}}(H,A,B)(r)>\lambda f_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function of the group $H$. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

Let $G$ be a finite simple graph and $J(G)$ denote its vertex cover ideal in a polynomial ring over a field $𝕂$. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset $S$ of the vertices of $G$ so that all symbolic powers of vertex cover ideals of $G\cup W(S)$, obtained from $G$ by adding a whisker to each vertex in $S$, have linear quotients. For instance, if $S$ is a vertex cover of $G$, then all symbolic powers of $J(G\cup W(S))$ have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose dimension of complementary space is maximal is a derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have $1$ dimension complementary space, admit local derivations which are not derivations. Moreover, similar problem concerning $2$-local derivations of such algebras is investigated and an example of solvable Leibniz algebra is given such that any $2$-local derivation on it is a derivation, but which admits local derivations which are not derivations.

This paper characterizes the potential behaviors of higher commutators in a simple algebra.

Higman’s group $H=〈a,b,c,d|b^a=b^2,c^b=c^2,d^c=d^2,a^d=a^2〉$ is a remarkable group with large (non-elementary) Dehn function. Higman constructed the group in 1951 to produce the first examples of infinite simple groups. Using finite state automata, and studying fixed points of certain finite state transducers, we show the conjugacy problem in $H$ is decidable for all inputs. Diekert, Laun and Ushakov have recently shown the word problem in $H$ is solvable in polynomial time, using the power circuit technology of Myasnikov, Ushakov and Won. Building on this work, we also show in a strongly generic setting that the conjugacy problem for $H$ has a polynomial time solution.

Let $F$ be a field of characteristic zero, algebraically closed and $E$ be the unitary infinite-dimensional Grassmann $F$-algebra. Our aim is to determine the minimal degree of an $A$-polynomial satisfied by $E\otimes E$. Also, we find an explicit $A$-identity of $E\otimes E$.

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^{\ast }$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever $𝜃$ is a ∧-preaction of a discrete inverse semigroupoid $S$ on an ample (possibly non-Hausdorff) groupoid ${𝒢}$, the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of ${𝒢}$ by $S$. This is a far-reaching generalization of analogous results which had been proven in particular cases.

Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös–Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about the homology of the nerve of a random Coxeter group and results about when random Coxeter groups are $\delta$-hyperbolic and when they have the FC-type property.