arXiv subjects
Publications and source records attributed to Alex Levine.
We show that, for every positive prime number $p$, the Baumslag-Gersten group $G_p = \langle a, t \mid (tat^{-1}) a(tat^{-1})^{-1} = a^p \rangle$ has an undecidable Diophantine problem. This gives the first known examples of one-relator groups with undecidable Diophantine problems. Furthermore we apply this to show that the one-relator group $G_p \ast \mathbb{Z}$ has an undecidable single equation problem.
Motivated by the open problem of whether all one-relator groups have decidable Diophantine problem, in this paper we prove a collection of undecidability results about the Diophantine problem for several families of groups that are close to one-relator groups in various ways. We prove that there is a generalised Baumslag--Solitar group with an undecidable Diophantine problem. Using our example we show there is a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group. Also, we prove that there is a one-relator product of cyclic groups with an undecidable Diophantine problem. In addition, we show that there there is a one-relator group $G$, with a single fixed finite rank free subgroup $H$, such that the Diophantine problem for $G$ with $H$-constraints is undecidable. The related open question of whether there is a free-by-cyclic group with undecidable Diophantine problem is also discussed, and we prove that there is a free-by-free group of the form $F_3 \rtimes F_2$ with an undecidable Diophantine problem.
The commutative subgroup width of a group $G$ is the smallest $k$ such that there are abelian subgroups $A_0,A_1,\ldots,A_{k-1}\leq G$ with $G=A_0A_1\cdots A_{k-1}$. Commutative (inverse) submonoid width is defined analogously. In 2002, Ab\'{e}rt showed, rather surprisingly, that the commutative subgroup width of the symmetric group on an infinite set is always finite. It was later shown by Seress that it is always bounded above by $14$. We answer a question of Seress and show that in fact the commutative subgroup width of $\operatorname{Sym}(\mathbb{N})$ is at most $9$. We improve the best known lower bound to $4$. We also study standard monoid analogues of the symmetric group; showing that the commutative submonoid widths of the full transformation monoid $\mathbb{N}^\mathbb{N}$, the partial transformation monoid $P_\mathbb{N}$ and the symmetric inverse monoid $I_\mathbb{N}$ are exactly $3$. We conclude by showing that the commutative inverse submonoid width of any infinite symmetric inverse monoid is always infinite.
We study an analogue of the conjugacy growth function in finitely generated groups: the automorphic growth function. This counts the number of automorphic orbits that intersect the ball of radius $n$ in the group. We show that this is not a commensurability invariant, by giving virtually abelian counterexamples. We classify the automorphic growth rate of all virtually abelian groups of rank at most $2$, the Heisenberg group, finite rank free groups and Thompson's groups $T$ and $V$. This last computation allows to conclude that $T$ and $V$ have exponential conjugacy growth.
In [arXiv:1405.6274, Question 5.2 & Question 5.3] Aschenbrenner, Friedl and Wilton ask: (1) Is the equation problem solvable for the fundamental group of any $3$-manifold? and (2) Is the first-order theory of the fundamental group of any $3$-manifold decidable? In this paper we answer both of these questions by proving that Hilbert's tenth problem over the integers can be encoded in equations over any non-virtually abelian fundamental group of any Seifert fibered 3-manifold whose orbifold has non-negative Euler characteristic. We use this to show that the equation problem (and hence also the first-order theory) is undecidable in this infinite family of $3$-manifold groups and then apply it to classify the Seifert 3-manifold groups with decidable equation problems and decidable first-order theories, in the case that the orbifold has non-negative Euler characteristic. In contrast, we show that for this class of Seifert 3-manifold groups the single equation problem is decidable. For every Seifert 3-manifold group $G$ where the orbifold has negative Euler characteristic we show that either $G$ has decidable equation problem or $G$ has a finite index subgroup of index $2$ that has decidable equation problem. These negative Euler characteristic results follow from work of Liang on central extensions of hyperbolic groups. We also discuss why Liang's results do not suffice to deal with all the negative Euler characteristic cases. We show how to construct several other infinite families of $3$-manifold groups with undecidable equation problem (and hence also undecidable first-order theory) including examples that are not Seifert manifold groups and examples that are not virtually nilpotent. In addition, we observe that there are numerous other infinite families for which the first-order theory is undecidable such as fundamental groups of manifolds modeled on 3-dimensional Sol geometry.
We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word problem is invariant under change of generating set and passing to finitely generated subgroups. This represents significant progress towards the conjecture that all groups with EDT0L word problem are finite (i.e. precisely the groups with regular word problem).
We show that the Diophantine problem in Thompson's group F is undecidable. Our proof uses the facts that F has finite commutator width and rank 2 abelianisation, then uses similar arguments used by B\"uchi and Senger and Ciobanu and Garreta to show the Diophantine problem in free groups and monoids with abelianisation constraints is undecidable.
In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from the theories of groups, automata, and inverse semigroups. An initial implementation of this algorithm outperforms existing implementations by several orders of magnitude.
In this paper we provide an overview of the class of inverse semigroups $S$ such that every congruence on $S$ relates at least one idempotent to a non-idempotent; such inverse semigroups are called $E$-disjunctive. This overview includes the study of the inverse semigroup theoretic structure of $E$-disjunctive semigroups; a large number of natural examples; some asymptotic results establishing the rarity of such inverse semigroups; and a general structure theorem for all inverse semigroups where the building blocks are $E$-disjunctive.
We study subsets $E$ of finitely generated groups where the set of all words over a given finite generating set that lie in $E$ forms a context-free language. We call these sets recognisably context-free. They are invariant of the choice of generating set and a theorem of Muller and Schupp fully classifies when the set $\{1\}$ can be recognisably context-free. We extend Muller and Schupp's result to show that a group $G$ admits a finite recognisably context-free subset if and only if $G$ is virtually free. We show that every conjugacy class of a group $G$ is recognisably context-free if and only if $G$ is virtually free. We conclude by showing that a coset is recognisably context-free if and only if the Schreier coset graph of the corresponding subgroup is quasi-isometric to a tree.
In this paper we study the satisfiability and solutions of group equations when combinatorial, algebraic and language-theoretic constraints are imposed on the solutions. We show that the solutions to equations with length, lexicographic order, abelianisation or context-free constraints added, can be effectively produced in finitely generated virtually abelian groups. Crucially, we translate each of the constraints above into a rational set in an effective way, and so reduce each problem to solving equations with rational constraints, which is decidable and well understood in virtually abelian groups. A byproduct of our results is that the growth series of a virtually abelian group, with respect to any generating set and any weight, is effectively computable. This series is known to be rational by a result of Benson, but his proof is non-constructive.
The survey provides an overview of the work done in the last 10 years to characterise solutions to equations in groups in terms of formal languages. We begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to systems of equations in free groups in terms of reduced words are expressible as EDT0L languages. We provide a sketch of their algorithm, and describe how the free group results extend to hyperbolic groups. The characterisation of solutions as EDT0L languages is very robust, and many group constructions preserve this, as shown by Levine. The most recent progress in the area has been made for groups without negative curvature, such as virtually abelian, the integral Heisenberg group, or the soluble Baumslag-Solitar groups, where the approaches to describing the solutions are different from the negative curvature groups. In virtually abelian groups the solutions sets are in fact rational, and one can obtain them as $m$-regular sets. In the Heisenberg group producing the solutions to a single equation reduces to understanding the solutions to quadratic Diophantine equations and uses number theoretic techniques. In the Baumslag-Solitar groups the methods are combinatorial, and focus on the interplay of normal forms to solve particular classes of equations. In conclusion, EDT0L languages give an effective and simple combinatorial characterisation of sets of seemingly high complexity in many important classes of groups.
Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms $g, h\colon\Sigma^*\to\Delta^*$ there exists any non-trivial $x\in\Sigma^*$ such that $g(x)=h(x)$. Post's Correspondence Problem for a group $\Gamma$ takes pairs of group homomorphisms $g, h\colon F(\Sigma)\to \Gamma$ instead, and similarly asks whether there exists an $x$ such that $g(x)=h(x)$ holds for non-elementary reasons. The restrictions imposed on $x$ in order to get non-elementary solutions lead to several interpretations of the problem; we consider the natural restriction asking that $x \notin \ker(g) \cap \ker(h)$ and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic $\Gamma$, but decidable when $\Gamma$ is virtually nilpotent. We also study this problem for group constructions such as subgroups, direct products and finite extensions. This problem is equivalent to an interpretation due to Myasnikov, Nikolev and Ushakov when one map is injective.
We express the solutions to quadratic equations with two variables in the ring of integers using EDT0L languages. We use this to show that EDT0L languages can be used to describe the solutions to one-variable equations in the Heisenberg group. This is done by reducing the question of solving a one-variable equation in the Heisenberg group to solving an equation in the ring of integers, exploiting the strong link between the ring of integers and nilpotent groups.
We show that the class of groups where EDT0L languages can be used to describe solution sets to systems of equations is closed under direct products, wreath products with finite groups, and passing to finite index subgroups. We also add the class of groups that contain a direct product of hyperbolic groups as a finite index subgroup to the list of groups where solutions to systems of equations can be expressed as an EDT0L language. This includes dihedral Artin groups. We also show that the systems of equations with rational constraints in virtually abelian groups have EDT0L solutions, and the addition of recognisable contraints to any system preserves the property of having EDT0L solutions. These EDT0L solutions are expressed with respect to quasigeodesic normal forms.
We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of Duchin, Liang and Shapiro to finite extensions.
This paper explores the nature of the solution sets of systems of equations in virtually abelian groups. We view this question from two angles. From a formal language perspective, we prove that the set of solutions to a system of equations forms an EDT0L language, with respect to a natural normal form. Looking at growth, we show that the growth series of the language of solutions is rational. Furthermore, considering the set of solutions as a set of tuples of group elements, we show that it has rational relative growth series with respect to any finite generating set.
The spontaneously oscillating hair bundle of sensory cells in the inner ear is an example of a stochastic, nonlinear oscillator driven by internal active processes. Moreover, this internal activity is adaptive -- its power input depends on the current state of the system. We study fluctuation dissipation relations in such adaptively-driven, nonequilibrium limit-cycle oscillators. We observe the expected violation of the well-known, equilibrium fluctuation-dissipation theorem (FDT), and verify the existence of a generalized fluctuation-dissipation theorem (GFDT) in the non-adaptively driven model of the hair cell oscillator. This generalized fluctuation theorem requires the system to be analyzed in the co-moving frame associated with the mean limit cycle of the stochastic oscillator. We then demonstrate, via numerical simulations and analytic calculations, that the adaptively-driven dynamical hair cell model violates both the FDT and the GFDT. We go on to show, using stochastic, finite-state, dynamical models, that such a feedback-controlled drive in stochastic limit cycle oscillators generically violates both the FDT and GFDT. We propose that one may in fact use the breakdown of the GFDT as a tool to more broadly look for and quantify the effect of adaptive, feedback mechanisms associated with driven (nonequilibrium) biological dynamics.