arXiv subjects

Murray Elder

Publications and source records attributed to Murray Elder.

At least 37 records · Page 2Linked to original sources

arXiv2020

The complexity of solution sets to equations in hyperbolic groups

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE$(n^2\log n)$ for the torsion-free case and NSPACE$(n^4\log n)$ in the torsion case. Furthermore, in the presence of quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L. Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Jeż, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis.

math.GR↗

arXiv2019

$k$-pop stack sortable permutations and $2$-avoidance

We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $k\in\mathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Guðmundsson. Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called \emph{$2$-avoidance}.

math.CO↗

arXiv2019

Solutions sets to systems of equations in hyperbolic groups are EDT0L in PSPACE

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, with or without torsion, as shortlex geodesic words, is an EDT0L language whose specification can be computed in $\mathsf{NSPACE}(n^2\log n)$ for the torsion-free case and $\mathsf{NSPACE}(n^4\log n)$ in the torsion case. Our work combines deep geometric results by Rips, Sela, Dahmani and Guirardel on decidability of existential theories of hyperbolic groups, work of computer scientists including Plandowski, Jeż, Diekert and others on $\mathsf{PSPACE}$ algorithms to solve equations in free monoids and groups using compression, and an intricate language-theoretic analysis. The present work gives an essentially optimal formal language description for all solutions in all hyperbolic groups, and an explicit and surprising low space complexity to compute them.

math.GR↗

arXiv2018

Bounded Automata Groups are co-ET0L

Holt and Röver proved that finitely generated bounded automata groups have indexed co-word problem. Here we sharpen this result to show they are in fact co-ET0L.

math.GR↗

arXiv2018

Applications of L systems to group theory

L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives in the free group on two generators, the Bridson-Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth.

math.GR↗

arXiv2018

Permutations sorted by a finite and an infinite stack in series

We prove that the set of permutations sorted by a stack of depth $t \geq 3$ and an infinite stack in series has infinite basis, by constructing an infinite antichain. This answers an open question on identifying the point at which, in a sorting process with two stacks in series, the basis changes from finite to infinite.

math.CO↗

arXiv2017

On a theorem of Avez

For each symmetric, aperiodic probability measure $μ$ on a finitely generated group $G$, we define a subset $A_μ$ consisting of group elements $g$ for which the limit of the ratio ${μ^{\ast n}(g)}/{μ^{\ast n}(e)}$ tends to $1$. We prove that $A_μ$ is a subgroup, is amenable, contains every finite normal subgroup, and $G=A_μ$ if and only if $G$ is amenable. For non-amenable groups we show that $A_μ$ is not always a normal subgroup, and can depend on the measure. We formulate some conjectures relating $A_μ$ to the amenable radical.

math.GR↗

arXiv2016

Sub-dominant cogrowth behaviour and the viability of deciding amenability numerically

We critically analyse a recent numerical method due to the first author, Rechnitzer and van Rensburg, which attempts to detect amenability or non-amenability in a finitely generated group by numerically estimating its asymptotic cogrowth rate. We identify two potential sources of error. We then propose a modification of the method that enables it to easily compute surprisingly accurate estimates for initial terms of the cogrowth sequence.

math.GR↗

arXiv2016

Permutations of context-free, ET0L and indexed languages

For a language $L$, we consider its cyclic closure, and more generally the language $C^k(L)$, which consists of all words obtained by partitioning words from $L$ into $k$ factors and permuting them. We prove that the classes of ET0L and EDT0L languages are closed under the operators $C^k$. This both sharpens and generalises Brandstädt's result that if $L$ is context-free then $C^k(L)$ is context-sensitive and not context-free in general for $k\geq 3$. We also show that the cyclic closure of an indexed language is indexed.

cs.FL↗

arXiv2016

Solution sets for equations over free groups are EDT0L languages

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jeż, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $\mathsf{NSPACE}(n\log n)$ here.

math.GR↗

arXiv2015

Solution sets for equations over free groups are EDT0L languages -- ICALP 2015 version

We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been been open for some years, apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products $\star_{1 \leq i \leq s}F_i$, where each $F_i$ is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by Ferté, Marin and Sénizergues (ToCS 2014), which is a very restricted case. Our general result became possible due to the recent recompression technique of Jeż. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper (arXiv 1405.5133). For example, we improve the complexity from quadratic nondeterministic space in (arXiv 1405.5133) to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: NSPACE($n\log n$). The conjectured complexity is NP, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured NP.

cs.LO↗

arXiv2015

Thompson's group F is 1-counter graph automatic

It is not known whether Thompson's group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to C-graph automatic by the authors, a compelling question is whether F is graph automatic or C-graph automatic for an appropriate language class C. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements.

math.GR↗

arXiv2015

Permutations of context-free and indexed languages

We consider the cyclic closure of a language, and its generalisation to the operators $C^k$ introduced by Brandstädt. We prove that the cyclic closure of an indexed language is indexed, and that if $L$ is a context-free language then $C^k(L)$ is indexed.

cs.FL↗

arXiv2014

2-manifold recognition is in logspace

We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.

math.GT↗

arXiv2014

Simple groups of automorphisms of trees determined by their actions on finite subtrees

We introduce the notion of the $k$-closure of a group of automorphisms of a locally finite tree, and give several examples of the construction. We show that the $k$-closure satisfies a new property of automorphism groups of trees that generalises Tits' Property $P$. We prove that, apart from some degenerate cases, any non-discrete group acting on a tree with this property contains an abstractly simple subgroup.

math.GR↗

arXiv2014

Metric properties of Baumslag-Solitar groups

We compute estimates for the word metric of Baumslag--Solitar groups in terms of the Britton's lemma normal form. As a corollary, we find lower bounds for the growth rate for the groups $BS(p,q)$, with $1<p\le q$.

math.GR↗

arXiv2014

Permutations generated by a depth 2 and infinite stack in series are algebraic

We prove that the class of permutations generated by passing an ordered sequence $12\dots n$ through a stack of depth 2 and an infinite stack in series is in bijection with an unambiguous context-free language, where a permutation of length $n$ is encoded by a string of length $3n$. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free language to compute the generating function: \begin{align*} \sum_{n\geq 0} c_n t^n &= \frac{(1+q)\left(1+5q-q^2-q^3-(1-q)\sqrt{(1-q^2)(1-4q-q^2)}\right)}{8q} \end{align*} where $c_n$ is the number of permutations of length $n$ that can be generated, and $q \equiv q(t) = \frac{1-2t-\sqrt{1-4t}}{2t}$ is a simple variant of the Catalan generating function. This in turn implies that $c_n^{1/n} \to 2+2\sqrt{5}$.

math.CO↗

arXiv2014

Minimal Almost Convexity

In this article we show that the Baumslag-Solitar group $BS(1,2)$ is minimally almost convex, or $MAC$. We also show that $BS(1,2)$ does not satisfy Poénaru's almost convexity condition $P(2)$, and hence the condition $P(2)$ is strictly stronger than $MAC$. Finally, we show that the groups $BS(1,q)$ for $q \geq 7$ and Stallings' non-$FP_3$ group do not satisfy $MAC$. As a consequence, the condition $MAC$ is not a commensurability invariant.

math.GR↗