Narrow Results

An M-automaton is a finite automaton with a blind counter that mimics a monoid M. The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M-automata play a central role in understanding the family 𝔏(M) of all languages accepted by M-automata. If G₁ and G₂ are finitely generated groups whose word problems are languages in 𝔏(M), in general, the word problem of the free product G₁ * G₂ is not necessarily in 𝔏(M). However, we show that if M is enlarged to the free product M*P₂, where P₂ is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M₁ * M₂ lies in 𝔏(M * P₂) whenever M₁ and M₂ are finitely generated monoids with special word problems in 𝔏(M * P₂). We also observe that there is a monoid without zero, denoted by CF₂, that can be used in place of P₂ for this purpose. The monoid CF₂ is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P₂. The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P₂ in the above result and under what circumstances P₁ (or the bicyclic monoid) is enough to do the job of P₂.

An unambiguous construction of an embedding of the free product of n cyclic groups of order p into the the group of finite state infinite unitriangular matrices over the finite field of size p is presented.

Let be a nonempty class of finite groups closed under taking subgroups, quotients and extensions. We consider groups G endowed with their pro- topology, and say that G is 2-subgroup separable if whenever H and K are finitely generated closed subgroups of G, then the subset HK is closed. We prove that if the groups G₁ and G₂ are 2-subgroup separable, then so is their free product G₁*G₂. This extends a result to T. Coulbois. The proof uses actions of groups on abstract and profinite trees.

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-cyclic. In general, the amalgamated product of free groups with cyclic amalgamations is free-by-(torsion-free nilpotent). The special case of the double of a free group amalgamating a cyclic subgroup is shown to be virtually free-by-abelian. Analagous results are obtained for certain one-relator HNN-extensions.

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. We prove that minor excluded graphs have finite Assouad–Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators, like ℤ³; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

We solve the subgroup intersection problem (SIP) for any RAAG $G$ of Droms type (i.e. with defining graph not containing induced squares or paths of length $3$): there is an algorithm which, given finite sets of generators for two subgroups $H,K\le G$, decides whether $H\cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H\cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. ${𝔽}_2\times {𝔽}_2$) even have unsolvable SIP.

We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$, where $n\ge 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$ admits an amalgamated free product structure.

The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(ℝ)$ of $n$ labeled points in $ℝ$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

A notion of "quasi-universal product" for algebraic probability spaces is introduced as a generalization of Speicher's "universal product". It is proved that there exist only five quasi-universal products, namely, tensor product, free product, Boolean product, monotone product and anti-monotone product. This result means that, in a sense, there exist only five independences which have nice properties of "associativity" and "(quasi-)universality".

Let be the class of all algebraic probability spaces. A "natural product" is, by definition, a map which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

We introduce and study a notion of Λ-freeness, which generalises both freeness and independence in the context of noncommutative probability. In particular, we extend Voiculescu's construction of the free product of representations of unital *-algebras. A central limit theorem is also proved.

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Schürmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behavior of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Schürmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of non-crossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Schürmann’s aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively, Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature.

In this paper, we introduce the new construction of quandles. For a group $G$ and a subset $A$ of $G$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that if $Q$ is a quandle such that the natural map $Q\to G_Q$ from $Q$ to the enveloping group $G_Q$ of $Q$ is injective, then $Q$ is the $(G,A)$-quandle for an appropriate group $G$ and a subset $A$ of $G$. Also we introduce the free product of quandles and study this construction for $(G,A)$-quandles. In addition, we classify all finite quandles with enveloping group ${\mathbb{Z}}^2$.

Let $P({\mathrm{\text{sl}}}_2(K))$ be the Poisson enveloping algebra of the Lie algebra ${\mathrm{\text{sl}}}_2(K)$ over an algebraically closed field $K$ of characteristic zero. The quotient algebras $P({\mathrm{\text{sl}}}_2(K))/(C_P-\lambda )$, where $C_P$ is the standard Casimir element of ${\mathrm{\text{sl}}}_2(K)$ in $P({\mathrm{\text{sl}}}_2(K))$ and $0\ne \lambda \in K$, are proven to be simple in [U. Umirbaev and V. Zhelyabin, A Dixmier theorem for Poisson enveloping algebras, J. Algebra 568 (2021) 576–600]. Using a result by Makar–Limanov [22], we describe generators of the automorphism group of $P({\mathrm{\text{sl}}}_2(K))/(C_P-\lambda )$ and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by Dixmier [Quotients simples de l’algebre enveloppante de ${𝔰𝔩}_2$, J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de $\mathrm{\text{Aut}}U({𝔰𝔩}_2)$ et $\mathrm{\text{Aut}}U(𝔥)$, J. Algebra 200 (1998) 404–427] for the quotient algebras $U({\mathrm{\text{sl}}}_2(K))/(C_U-\lambda )$, where $C_U$ is the standard Casimir element of ${\mathrm{\text{sl}}}_2(K)$ in the universal enveloping algebra $U({\mathrm{\text{sl}}}_2(K))$, we prove that the automorphism groups of $P({\mathrm{\text{sl}}}_2(K))/(C_P-\lambda )$ and $U({\mathrm{\text{sl}}}_2(K))/(C_U-\lambda )$ are isomorphic.

The groupoid description of Schwinger’s picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger’s groupoid of the system. The groupoid picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, as well as the notion of free independence, find a natural realm in the groupoid formalism. The ideas described in this paper will be illustrated by using the EPRB experiment. It will be observed that, in addition to the notion of the non-separability provided by the entangled state of the system, there is an intrinsic “non-separability” associated to the impossibility of identifying the entangled particles as subsystems of the total system.

We apply the method of Gröbner–Shirshov bases for replicated algebras developed by Kolesnikov to offer a general approach for constructing free products of associative trialgebras (or trioids). In particular, the open problem of Zhuchok on constructing free products of trioids is solved.