Length-preserving geodesic rewriting for Artin groups with $A_3$ or $B_3$ subdiagrams: the rank-three types $(2,3,3)$ and $(2,3,4)$

For the rank-three Artin groups $G(2,3,n)=\langle a,b,c \mid aba=bab,\ ac=ca,\ (b,c)_n=(c,b)_n\rangle$ , where $(x,y)_n$ denotes the alternating word of length $n$ beginning with $x$ , Holt and Rees (2023) proved that for $n\ge5$ the word problem is solved by a system of length-preserving rewrite rules which, together with free reduction, reduces any word to a geodesic representative in quadratic time, with any two geodesics of the same element connected by two-generator moves. A December 2024 continuation by Blasco-García, Cumplido, Holt, Morris-Wright and Rees extended this to all Artin groups whose Coxeter diagram contains no $A_3$ or $B_3$ subdiagram, and both papers show by explicit example that the exclusion is necessary for the current critical-word/rightward-reducing-sequence method, which fails at $n=4$ . The open problem is to settle the excluded cases, the smallest being the rank-three groups of type $(2,3,3)$ (spherical type $A_3$ ) and $(2,3,4)$ (spherical type $B_3$ ): either construct an analogous length-preserving rewrite system reducing every word to a geodesic in quadratic time, with the corresponding two-generator geodesic-connectivity property, or prove that no such length-preserving rewriting scheme exists. The complexity side is not at issue: these spherical-type groups have long had quadratic-time word problems via Garside theory, and Haettel–Huang independently established quadratic Dehn functions for all rank-three Artin groups.

Background

An Artin group is a group defined by a presentation with one generator per node of a labelled graph (the Coxeter diagram, the labels being the orders $m_{x,y}$ of the images of the generators in an associated Coxeter group), in which each pair of standard generators $x,y$ satisfies one braid-type relation: the alternating word of length $m_{x,y}$ beginning with $x$ equals the alternating word of length $m_{x,y}$ beginning with $y$ . A label $m_{x,y}=2$ means the two generators simply commute. Holt–Rees, 'Artin groups of type (2,3,n)' (arXiv:2310.14632), studies the three-generator case

$G(2,3,n)=\langle a,b,c \mid aba=bab,\ ac=ca,\ (b,c)_n=(c,b)_n\rangle,$

where $(x,y)_n$ denotes the alternating word $xyxy\cdots$ of length $n$ beginning with $x$ ; here $a$ and $b$ satisfy the length-3 braid relation, $a$ and $c$ commute, and $b$ and $c$ satisfy a braid relation of length $n$ . For $n=3,4,5$ the associated Coxeter group is finite (the diagram is of spherical type $A_3$ , $B_3$ , $H_3$ respectively), while for $n\ge6$ the group is non-spherical and two-dimensional, hence locally non-spherical in Chermak's sense (no standard subgroup generated by three of the generators is finite — for a rank-three group, this only says the group itself is infinite).

A word over the generators and their inverses is geodesic if no shorter word represents the same group element, and the word problem asks for an algorithm deciding whether a word represents the identity. The line of work considered here seeks solutions of a specific structural kind: a finite or finitely-described system of length-preserving rewrite rules (rules replacing a subword by an equal-length word representing the same group element) which, together with free reduction (cancellation of adjacent inverse letters $xx^{-1}$ ), reduces any input word to a geodesic representative in time quadratic in the word length, and such that any two geodesic representatives of the same element are connected by a sequence of these rules, each involving just two generators. In the predecessor papers the rules are so-called $\tau$ -moves: a critical subword $u$ (an alternating word of maximal length in two generators, or a 'pseudo-2-generated' variant thereof, or in Holt–Rees a new 'pseudo-3-generated' type built from two overlapping two-generator critical words) is replaced by an equal-length word $\tau(u)$ whose first and last letters differ from those of $u$ . Sequences of $\tau$ -moves that progress from left to right through the word and end in a free cancellation are called rightward reducing sequences (RRS), and the main technical content is proving that every non-geodesic word admits one.

Prior results framing the problem: (1) Spherical-type Artin groups have solvable word problems via Garside theory — a Garside group is a group admitting a lattice-like structure generalizing braid groups, yielding normal forms and a quadratic-time word problem (Dehornoy–Paris 1999) — but this does not produce length-preserving rewrite systems reducing words to geodesics. (2) Chermak (1998) gave an algorithm for locally non-spherical Artin groups that reduces any word representing the identity to the empty word without ever increasing length, but does not reduce an arbitrary word to a geodesic. (3) Holt–Rees (2012) proved that large-type Artin groups (all labels $\ge3$ ) are shortlex automatic with regular geodesics — that is, both the set of geodesic words and the language of shortlex representatives (words ordered first by length, then alphabetically) are recognized by finite automata, which yields a quadratic-time solution of the word problem. (4) Holt–Rees (arXiv:2310.14632, 2023) covered the remaining rank-three groups $G(2,3,n)$ for $n\ge5$ , and observed that the same arguments apply when the label 3 is replaced by any label greater than 3. (5) Haettel–Huang (Duke Math. J. 174, 2025) proved that $G\times\mathbb{Z}$ is a Garside group for large-type Artin groups of rank at most 3 and a broader class, which by their Theorem D implies biautomaticity (the automatic-structure property of item (3), strengthened so that the representative language is also stable under multiplication by generators on the other side), solvable word and conjugacy problems, and quadratic Dehn function (the Dehn function measures the area needed to fill closed loops in the Cayley complex, and bounds word-problem complexity) for every rank-three Artin group — so asymptotic solvability is not what is at stake; the open question concerns the specific combinatorial rewriting solution and the structure of geodesics. The structure of geodesics in rank-three Artin groups is also an ingredient in the study of the Artin monoid Cayley graph (the Cayley graph taken with respect to the Artin monoid, i.e. the monoid on the same generators and braid relations but without inverses) by Boyd–Charney–Morris-Wright–Rees, where the groups $G(2,3,n)$ , $n\ge6$ , were the rank-three infinite-type cases not covered by the word-problem solutions available to those authors. Finally, the direct continuation Blasco-García–Cumplido–Holt–Morris-Wright–Rees, 'Rewriting in Artin groups without $A_3$ or $B_3$ subdiagrams' (arXiv:2412.12195, December 2024), extended the theorem to arbitrary rank for diagrams containing no $A_3$ or $B_3$ subdiagram — that is, no generators $x,y,z$ with $xyx=yxy$ , $xz=zx$ , and $(y,z)_n$ -relations with $n=3$ or $n=4$ — while proving by example that this exclusion is necessary for the current method: in the $n=4$ group the word $(bab)(cbca)(bc)b^{-1}$ is reduced to a shorter word by the scheme, yet the word $bacbcbabcb^{-1}$ , representing the same element, admits no rightward reducing sequence at all (its only critical subwords are $cbcb$ and $bab$ , and neither can serve as the first move of such a sequence). The class containing such subdiagrams — whose smallest instances are exactly the rank-three groups $G(2,3,3)$ and $G(2,3,4)$ — is therefore the precise residue left open by this line of work.

Problem Statement

For the rank-three Artin groups

$G(2,3,n)=\langle a,b,c \mid aba=bab,\ ac=ca,\ (b,c)_n=(c,b)_n\rangle,\qquad n\in\{3,4\},$

where $(x,y)_n$ denotes the alternating word of length $n$ beginning with $x$ — and, more generally, for Artin groups whose Coxeter diagram contains an $A_3$ or $B_3$ subdiagram, the exact class excluded by the theorem of Blasco-García–Cumplido–Holt–Morris-Wright–Rees (arXiv:2412.12195) — determine whether the word problem admits a solution of the Holt–Rees type: a system of length-preserving rewrite rules over the standard generators which, together with free reduction (cancellation of adjacent inverse letters), reduces every word to a geodesic representative of the same group element in time quadratic in the word length, with the additional property that any two geodesic representatives of the same element are connected by a sequence of these rules, each involving at most two generators. A positive resolution would be such an explicit system, with proofs that its rules preserve length and hold as identities in the group, that the reduction procedure always terminates at a geodesic, that it runs in quadratic time, and that the geodesic-connectivity property holds; a negative resolution would be a rigorous proof that no system of length-preserving rewrite rules over the standard generators, together with free reduction, reduces every word to a geodesic representative of the same element in time quadratic in the word length, with the additional property that any two geodesic representatives of the same element are connected by a sequence of these rules, each involving at most two generators. The documented failure at $n=4$ of the specific critical-word/rightward-reducing-sequence method of Holt–Rees (Example 4.5 of arXiv:2310.14632; Example 4.6 of arXiv:2412.12195) does not by itself constitute a negative resolution, since it establishes failure only for that one method family; the negative resolution concerns length-preserving rewriting schemes for these groups generally, negating the Holt–Rees-type existence question above, whose reduction-only form is the general open question posed in arXiv:2412.12195 of whether every Artin group's word problem can always be solved using length-preserving rewrites together with free reduction. Any proposed system must overcome the documented obstruction at $n=4$ : the specific $\tau$ -move and rightward-reducing-sequence analysis of Holt–Rees fails there, since $(bab)(cbca)(bc)b^{-1}$ reduces under it while $bacbcbabcb^{-1}$ , representing the same element, admits no rightward reducing sequence.

References

Derek F. Holt, Sarah Rees, 'Artin groups of type (2,3,n)', 2023, arXiv:2310.14632, DOI 10.48550/arXiv.2310.14632, https://arxiv.org/abs/2310.14632
Rubén Blasco-García, María Cumplido, Derek F. Holt, Rose Morris-Wright, Sarah Rees, 'Rewriting in Artin groups without A3 or B3 subdiagrams', 2024, arXiv:2412.12195, DOI 10.48550/arXiv.2412.12195, https://arxiv.org/abs/2412.12195
Rubén Blasco-García, María Cumplido, Rose Morris-Wright, 'The Word Problem is Solvable for 3-free Artin groups in Quadratic Time', 2022, revised 2025 (v4), arXiv:2204.03523, DOI 10.48550/arXiv.2204.03523, https://arxiv.org/abs/2204.03523
Thomas Haettel, Jingyin Huang, 'New Garside structures and applications to Artin groups', Duke Mathematical Journal 174 (9) (2025), DOI 10.1215/00127094-2024-0062, preprint arXiv:2305.11622, https://doi.org/10.1215/00127094-2024-0062
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A. Chermak, 'Locally non-spherical Artin groups', Journal of Algebra 200 (1998) 56–98, DOI 10.1006/jabr.1997.7229, https://doi.org/10.1006/jabr.1997.7229
P. Dehornoy, L. Paris, 'Gaussian groups and Garside groups, two generalisations of Artin groups', Proceedings of the London Mathematical Society (3) 79 (1999) 569–604, DOI 10.1112/s0024611599012071, https://doi.org/10.1112/s0024611599012071
Rachael Boyd, Ruth Charney, Rose Morris-Wright, Sarah Rees, 'The Artin monoid Cayley graph', Journal of Combinatorial Algebra 10 (1) (2024) 131–151, DOI 10.4171/jca/85, preprint arXiv:2303.09504, https://doi.org/10.4171/jca/85