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Derek F. Holt

Publications and source records attributed to Derek F. Holt.

At least 19 recordsLinked to original sources
arXiv2024

Rewriting in Artin groups without A_3 or B_3 subdiagrams

We prove that the word problem in an Artin group G based on a diagram without A_3 or B_3 subdiagrams can be solved using a system of length preserving rewrite rules which, together with free reduction, can be used to reduce any word over the standard generators of G to a geodesic word in G in quadratic time. This result builds on work of Holt and Rees, and of Blasco-Garc\'ia, Cumplido and Morris-Wright. Those articles prove the same result for all Artin groups that are either sufficiently large or 3-free, respectively.

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arXiv2024

Groups with ET0L co-word problem

We study groups whose co-word problems are ET0L languages, which we call coET0L groups, using an automaton based model due to van Leeuwen, and recently studied by Bishop and Elder. In particular we prove a number of closure results for the class of groups with co-word problems in a subclass of `special' ET0L languages; that class of groups contains all groups that we know at the time of writing to be co-ET0L, including all groups that were proved by Holt and R\"over to be stack groups, and hence co-indexed. It includes virtually free groups, bounded automata groups, and the Higman-Thompson groups, together with groups constructed from those using finitely generated subgroups, finite extension, free and direct products, and by taking the restricted standard wreath product of a co-\E group by a finitely generated virtually free top group.

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arXiv2024

Minimal sized generating sets of permutation groups

We present a randomised variant of an algorithm of Lucchini and Thakkar for finding a smallest sized generating set in a finite group, which has polynomial time expected running time in finite permutation groups.

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arXiv2023

Artin groups of type (2,3,n)

Our main theorem is that the word problem in the Artin group G = for n >= 5 can be solved using a system R of length preserving rewrite rules that, together with free reduction, can be used to reduce any word over {a,b,c} to a geodesic word in $G$, in quadratic time. This result builds on work of Holt and Rees, and of Blasco, Cumplido and Morris-Wright, which proves the same result for all Artin groups that are either sufficiently large or 3-free. Since every rank 3 Artin group is either spherical or in one of the categories covered by the previous results on which we build, it follows that any rank 3 Artin group has quadratic Dehn function. However we note that this and much more is a consequence of very recent work of Haettel and Huang; our contribution is to provide a particular kind or rewriting solution to the word problem for the non-spherical rank 3 Artin groups (and more).

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arXiv2023

Using EDT0L systems to solve some equations in the solvable Baumslag-Solitar groups

We investigate the solution sets to equations in the solvable Baumslag-Solitar groups $BS(1,k)$, $k\geq2$, and show that these sets are represented by EDT0L languages in some cases. In particular, we prove that the multiplication table of such a group forms an EDT0L language with respect to a specific natural normal form for group elements.

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arXiv2022

Isomorphism and non-isomorphism for interval groups of type D_n

We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentations for the groups, and use these new presentations to show that the interval group associated with a proper quasi-Coxeter element of $W(D_n)$ cannot be isomorphic to the Artin group of type $D_n$. While the specific problems we solve arise from the study of interval groups, their solution provides an illustration of how techniques indicated by computational observation can be used to derive properties of all groups within an infinite family.

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arXiv2022

Interval groups related to finite Coxeter groups, Part II

We provide a complete description of the presentations of the interval groups related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4-cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non-simply laced cases, where a new Garside structure is discovered. Furthermore, we obtain a complete classification of whether the interval group we consider is isomorphic or not to the related Artin group. Indeed, using methods of Tits, we prove that the interval groups of proper quasi-Coxeter elements are not isomorphic to the Artin groups of the same type, in the case of $D_n$ when $n$ is even or in any of the exceptional cases. In [BHNR22], we show using different methods that this result holds for type $D_n$ for all $n \geq 4$.

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arXiv2017

Multifraction reduction IV: Padding and Artin-Tits groups of sufficiently large type

We investigate the padded version of reduction, an extension of multifraction reduction as defined in arXiv:1606.08991, and connect it both with ordinary reduction and with the so-called Property $\mathrm{H}$. As an application, we show that all Artin-Tits groups of sufficiently large type satisfy some weakening Conjecture $\mathrm{A^{padded}}$ of Conjecture $\mathrm{A}$, thus showing that the reduction approach is relevant for these groups.

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arXiv2015

The generalised word problem for subgroups of hyperbolic groups

We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infinite index with context-free generalised word problem, and that, for any hyperbolic group, the generalised word problem of a torsion-free quasiconvex subgroup is recognised by a real-time Turing machine.

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arXiv2011

The conjugacy problem in hyperbolic groups for finite lists of group elements

Let G be a word-hyperbolic group with given finite generating set, for which various standard structures and constants have been pre-computed. A (non-practical) algorithm is described that, given as input two lists A and B, each composed of m words in the generators and their inverses, determines whether or not the lists are conjugate in G, and returns a conjugating element should one exist. The algorithm runs in time O(m mu)$, where mu is an upper bound on the lengths of elements in the two lists. Similarly, an algorithm is outlined that computes generators of the centraliser of A, with the same bound on running time.

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arXiv2011

A characterisation of virtually free groups

We prove that a finitely generated group $G$ is virtually free if and only if there exists a generating set for $G$ and $k > 0$ such that all $k$-locally geodesic words with respect to that generating set are geodesic.

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arXiv2011

Groups whose geodesics are locally testable

A regular set of words is ($k$-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length $k$. In this article we study groups for which the set of all geodesic words with respect to some generating set is ($k$-)locally testable, and we call such groups ($k$-)locally testable. We show that a group is \klt{1} if and only if it is free abelian. We show that the class of ($k$-)locally testable groups is closed under taking finite direct products. We show also that a locally testable group has finitely many conjugacy classes of torsion elements. Our work involved computer investigations of specific groups, for which purpose we implemented an algorithm in \GAP\ to compute a finite state automaton with language equal to the set of all geodesics of a group (assuming that this language is regular), starting from a shortlex automatic structure. We provide a brief description of that algorithm.

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arXiv2011

Star-free geodesic languages for groups

In this article we show that every group with a finite presentation satisfying one or both of the small cancellation conditions $C'(1/6)$ and $C'(1/4)-T(4)$ has the property that the set of all geodesics (over the same generating set) is a star-free regular language. Star-free regularity of the geodesic set is shown to be dependent on the generating set chosen, even for free groups. We also show that the class of groups whose geodesic sets are star-free with respect to some generating set is closed under taking graph (and hence free and direct) products, and includes all virtually abelian groups.

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arXiv2011

Generalising some results about right-angled Artin groups to graph products of groups

We prove three results about the graph product $G=\G(Γ;G_v, v \in V(Γ))$ of groups $G_v$ over a graph $Γ$. The first result generalises a result of Servatius, Droms and Servatius, proved by them for right-angled Artin groups; we prove a necessary and sufficient condition on a finite graph $Γ$ for the kernel of the map from $G$ to the associated direct product to be free (one part of this result already follows from a result in S. Kim's Ph.D. thesis). The second result generalises a result of Hermiller and Sunic, again from right-angled Artin groups; we prove that for a graph $Γ$ with finite chromatic number, $G$ has a series in which every factor is a free product of vertex groups. The third result provides an alternative proof of a theorem due to Meier, which provides necessary and sufficient conditions on a finite graph $Γ$ for $G$ to be hyperbolic.

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arXiv2008

Groups that do and do not have context-sensitive word problem

We prove that a group has word problem that is a growing context-sensitive language precisely if its word problem can be solved using a non-deterministic Cannon's algorithm (the deterministic algorithms being defined by Goodman and Shapiro). We generalise their results to find many examples of groups not admitting non-deterministic Cannon's algorithms. This adds to the examples of Kambites and Otto of groups separating context-sensitive and growing context-sensitive word problems, and provides a new language-theoretic separation result.

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arXiv1999

Combing nilpotent and polycyclic groups

A combing is a set of normal forms for a finitely generated group. This article investigates the language-theoretic and geometric properties of combings for nilpotent and polycyclic groups. It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are also all 2 or 3-generated class 2 nilpotent groups, and groups in certain families of nilpotent groups, e.g. the finitely generated Heisenberg groups, groups of unipotent matrices over the integers and the free class 2 nilpotent groups. Further it is shown that any polycyclic-by-finite group embeds in a real-time combable group. All the combings constructed in the article are boundedly asynchronous, and those for nilpotent-by-finite groups have polynomially bounded length functions, of degree equal to the nilpotency class, c. This result verifies a polynomial upper bound on the Dehn functions of those groups of degree c+1.

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