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Derek Holt

Publications and source records attributed to Derek Holt.

15 recordsLinked to original sources
arXiv2026

Constructing free subgroups of matrix groups

The Tits alternative asserts that a finitely generated linear group is either solvable-by-finite or contains a non-abelian free subgroup. Given a non-solvable-by-finite subgroup G of GL(d, F), where F is the field of complex numbers or a non-archimedean local field, we present methods to construct explicitly a non-abelian 2-generator free subgroup of G in specific cases. Our implementations in the computeralgebra system Oscar are publicly available.

math.GR↗
arXiv2024

Compressed decision problems in hyperbolic groups

We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight line programs defined over a finite generating set for the group. We prove also that, for any infinite hyperbolic group $G$, the compressed knapsack problem in $G$ is ${\mathsf{NP}}$-complete.

math.GR↗
arXiv2021

The transitive groups of degree 48 and some applications

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195\,826\,352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$ of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive graphs of degree $48$, of which there are $1\,538\,868\,366$, all but $0.1625\%$ of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published by the third author as a separate paper

math.GR↗
arXiv2020

Polynomial-time proofs that groups are hyperbolic

It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings, to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure which analyses these diagrams, and either returns an explicit linear Dehn function for the presentation, or returns fail, together with its reasons for failure. Furthermore, if our procedure succeeds we are often able to produce in polynomial time a word problem solver for the presentation that runs in linear time. Our algorithms have been implemented, and are often many orders of magnitude faster than KBMAG, the only comparable publicly available software.

math.GR↗
arXiv2018

A Census of Small Transitive Groups and Vertex-Transitive Graphs

We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than $48$, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than $48$. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focussing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list of the elusive groups of order at most $47$ and then by completely determining which groups of order at most $47$ are CI groups

math.CO↗
arXiv2018

Biautomatic structures in systolic Artin groups

We examine the construction of Huang and Osajda that was used in their proof of the biautomaticity of Artin groups of almost large type. We describe a slightly simpler variant of that biautomatic structure, with explicit descriptions of a few small examples, and we examine some of the properties of the structure. We explain how the construction can be programmed within the GAP system.

math.GR↗
arXiv2016

The generalised word problem in hyperbolic and relatively hyperbolic groups

We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove 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.

math.GR↗
arXiv2014

Conjugacy languages in groups

We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.

math.GR↗
arXiv2013

Algorithms and topology for Cayley graphs of groups

Autostackability for finitely generated groups is defined via a topological property of the associated Cayley graph which can be encoded in a finite state automaton. Autostackable groups have solvable word problem and an effective inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. A comparison with automatic groups is given. Another characterization of autostackability is given in terms of prefix-rewriting systems. Every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix-closed set of normal forms is also autostackable. As a consequence, the fundamental group of every closed 3-manifold with any of the eight possible uniform geometries is autostackable.

math.GR↗