arXiv subjects

Carl-Fredrik Nyberg-Brodda

Publications and source records attributed to Carl-Fredrik Nyberg-Brodda.

At least 37 records · Page 2Linked to original sources

arXiv2024

The growth of free inverse monoids

We compute the rate of exponential growth of the free inverse monoid of rank $r$ (and hence an upper bound on the corresponding rate for all $r$-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to $\sqrt{e(2k-1)}$ as $k \to \infty$.

math.GR↗

arXiv2024

Membership problems for positive one-relator groups and one-relation monoids

Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form $w=1$ but which are not groups, and examples defined by relations of the form $u=v$ where both of $u$ and $v$ are non-empty. As a consequence we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form $aUb = a$, and examples of the form $aUb=aVa$, with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form $uv^{-1}$ with $u, v$ positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.

math.GR↗

arXiv2024

On congruence subgroups of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ generated by two parabolic elements

We study the freeness problem for subgroups of $\operatorname{SL}_2(\mathbb{C})$ generated by two parabolic matrices. For $q = r/p \in \mathbb{Q} \cap (0,4)$, where $p$ is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $Δ_q$ generated by the two matrices \[ A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \text{ and } Q_q = \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix}. \] We introduce the conjecture that $Δ_{r/p} = \overlineΓ_1^{(p)}(r)$, the congruence subgroup of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ consisting of all matrices with upper right entry congruent to $0$ mod $r$ and diagonal entries congruent to $1$ mod $r$. We prove this conjecture when $r \leq 4$ and for some cases when $r = 5$. Furthermore, conditional on a strong form of Artin's conjecture on primitive roots, we also prove the conjecture when $r \in \{ p-1, p+1, (p+1)/2 \}$. In all these cases, this gives information about the algebraic structure of $Δ_{r/p}$: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index $J_2(r)$ in $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$, where $J_2(r)$ denotes the Jordan totient function.

math.GR↗

arXiv2024

G. S. Tseytin's seven-relation semigroup with undecidable word problem

We give an introduction to the ideas behind G. S. Tseytin's 1958 construction of a seven-relation semigroup with undecidable word problem. We give a history of the ideas leading up to its construction, some intuition for the proof, and provide an overview of some subsequent results and developments which stem from this remarkable semigroup. An English translation of Tseytin's article by the author of the present article is supplemented at the end of the article (the Russian original was published in Trudy Mat. Inst. Steklov 52 (1958), pp. 172--189).

math.HO↗

arXiv2024

The abelianization of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{m}])$

For all $m \geq 1$, we prove that the abelianization of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{m}])$ is (1) trivial if $6 \mid m$; (2) $\mathbb{Z} / 3\mathbb{Z}$ if $2 \mid m$ and $\gcd(3,m)=1$; (3) $\mathbb{Z} / 4 \mathbb{Z}$ if $3 \mid m$ and $\gcd(2,m)=1$; and (4) $\mathbb{Z} / {12}\mathbb{Z} \cong \mathbb{Z} / 3\mathbb{Z} \times \mathbb{Z} / 4\mathbb{Z}$ if $\gcd(6,m)=1$. This completes known computational results of Bui Anh & Ellis for $m \leq 50$. The proof is completely elementary, and in particular does not use the congruence subgroup property. We also find a new presentation for $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{2}])$. This presentation has two generators and three relators. Thus, $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{2}])$ admits a presentation with deficiency equal to the rank of its Schur multiplier. This also gives new and very simple presentations for the finite groups $\operatorname{SL}_2(\mathbb{Z} / m \mathbb{Z})$, where $m$ is odd.

math.KT↗

arXiv2023

Six Ukrainian articles on semigroup theory by A. K. Sushkevich

This is an English translation of six articles, originally written in Ukrainian, by the semigroup theorist Anton Kazimirovich Sushkevich (1889-1961). The articles date between 1935 and 1939, and were all written in Kharkiv. A preface is also included, giving a summary of the articles and the purpose of the translation.

math.HO↗

arXiv2022

On the Dehn functions of a class of monadic one-relation monoids

We give an infinite family of monoids $Π_N$ (for $N=2, 3, \dots$), each with a single defining relation of the form $bUa = a$, such that the Dehn function of $Π_N$ is at least exponential. More precisely, we prove that the Dehn function $\partial_N(n)$ of $Π_N$ satisfies $\partial_N(n) \succeq N^{n/4}$. This answers negatively a question posed by Cain & Maltcev in 2013 on whether every monoid defined by a single relation of the form $bUa=a$ has quadratic Dehn function. Finally, by using the decidability of the rational subset membership problem in the metabelian Baumslag--Solitar groups $\operatorname{BS}(1,n)$ for all $n \geq 2$, proved recently by Cadilhac, Chistikov & Zetzsche, we show that each $Π_N$ has decidable word problem.

math.GR↗

arXiv2022

Tropical one-relation monoids

In 2010, Izhakian & Margolis proved that the bicyclic monoid admits a representation as a semigroup of upper triangular tropical matrices. We extend this result by classifying all one-relation monoids which admit such representations. We also classify, with two exceptions, all one-relation monoids which admit representations as semigroups of tropical matrices.

math.RA↗

arXiv2022

On the Diophantine problem in some one-relator groups

We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister--Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation $a^mb^n = 1$ is virtually a direct product of hyperbolic groups for all $m, n \geq 0$, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the two-generator, one-relator groups $G_{m,n}$ with defining relation a commutator $[a^m, b^n] = 1$, where $m, n \geq 1$. In doing so, we define and study a natural class of groups (RABSAGs), related to right-angled Artin groups (RAAGs). We reduce the Diophantine problem in the groups $G_{m,n}$ to the Diophantine problem in groups which are virtually certain RABSAGs. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group $G_{2,2}$ with the single defining relation $[a^2, b^2] = 1$. We use the recent classification by Gray & Howie of RAAG subgroups of one-relator groups to classify the RAAG subgroups of some RABSAGs, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups $\operatorname{NG}(p,q)$, which are $(p+1)$-generated one-relator groups generalising the solvable Baumslag--Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.

math.GR↗

arXiv2022

A word-hyperbolic special monoid with undecidable Diophantine problem

The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special monoid $M$ with trivial group of units, and such that the Diophantine problem is undecidable in $M$. This answers two questions asked by Garreta & Gray in 2019, and shows that the decidability of the Diophantine problem in hyperbolic groups, as proved by Dahmani & Guirardel, does not generalise to word-hyperbolic monoids.

math.GR↗

arXiv2022

Multiplication tables and word-hyperbolicity in free products of semigroups, monoids, and groups

This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied e.g. by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids, or groups with a $\mathbf{C}$-multiplication table, where $\mathbf{C}$ is any reversal-closed super-$\operatorname{AFL}$, in the sense of Greibach. In particular, we deduce that the free product of two groups with $\operatorname{ET0L}$ resp. indexed multiplication tables again has an $\operatorname{ET0L}$ resp. indexed multiplication table.

math.GR↗

arXiv2022

On the Word Problem for Compressible Monoids

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if $\mathcal{C}$ is a reversal-closed super-$\operatorname{AFL}$, as defined by Greibach, then $M$ has word problem in $\mathcal{C}$ if and only if its compressed left monoid $L(M)$ has word problem in $\mathcal{C}$. As a special case, we may take $\mathcal{C}$ to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalisation of a question first asked by Zhang in 1992.

math.GR↗

arXiv2021

On the Word Problem for Free Products of Semigroups and Monoids

We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which generalise e.g. the context-free or the indexed languages. When $\mathcal{C}$ is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in $\mathcal{C}$ also has word problem in $\mathcal{C}$. This recovers and generalises a recent result by Brough, Cain & Pfeiffer that the class of context-free semigroups (monoids) is closed under taking free products. As a group-theoretic corollary, we deduce that the word problem of the (group) free product of two groups with word problem in $\mathcal{C}$ is also in $\mathcal{C}$. As a particular case, we find that the free product of two groups with indexed word problem has indexed word problem.

math.GR↗

arXiv2021

On one-relator groups and units of special one-relation inverse monoids

This note investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse monoids, i.e. those inverse monoids with a presentation of the form $\operatorname{Inv}\langle A \mid w=1 \rangle$. We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes ${\rm {\small ANY}}, {\rm {\small RED}}, {\rm {\small CRED}},$ and ${\rm {\small POS}}$ of one-relator groups which can be defined by special one-relation inverse monoid presentations in which the defining word is arbitrary; reduced; cyclically reduced; or positive, respectively. We show that the inclusions ${\rm {\small ANY}} \supset {\rm {\small CRED}} \supset {\rm {\small POS}}$ are all strict. Conditional on a natural conjecture, we prove ${\rm {\small ANY}} \supset {\rm {\small RED}}$. Following this, we use the Benois algorithm recently devised by Gray & Ruskuc to produce an infinite family of special one-relation inverse monoids which exhibit similar pathological behaviour (which we term O'Haresque) to the O'Hare monoid with respect to computing the minimal invertible pieces of the defining word. Finally, we provide a counterexample to a conjecture by Gray & Ruskuc that the Benois algorithm always correctly computes the minimal invertible pieces of a special one-relation inverse monoid.

math.GR↗

arXiv2021

On the Word Problem for Special Monoids

A monoid is called special if it admits a presentation in which all defining relations are of the form $w = 1$. Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan & Gilman in 2004. We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang in 1992. As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.

math.GR↗

arXiv2021

Non-finitely Generated Maximal Subgroups of Context-free Monoids

A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups.

math.GR↗

arXiv2021

The Word Problem for One-relation Monoids: A Survey

This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem.

math.GR↗

arXiv2021

The B. B. Newman Spelling Theorem

This article aims to be a self-contained account of the history of the B. B. Newman Spelling Theorem, including the historical context in which it arose. First, an account of B. B. Newman and how he came to prove his Spelling Theorem is given, together with a description of the author's efforts to track this information down. Following this, a high-level description of combinatorial group theory is given. This is then tied in with a description of the history of the word problem, a fundamental problem in the area. After a description of some of the theory of one-relator groups, an important part of combinatorial group theory, the natural division line into the torsion and torsion-free case for such groups is described. This culminates in a statement of and general discussion about the B. B. Newman Spelling Theorem and its importance.

math.HO↗