The Word Problem is Solvable for 3-free Artin groups in Quadratic Time

TL;DR

The paper addresses the word problem for $3$-free Artin groups (i.e., Artin groups with no relations of length $3$) by developing a quadratic-time algorithm. It generalizes Holt–Rees’ dihedral techniques via Rightward Reducing Sequences (RRS) built from pseudo 2-generated (P2G) subwords and generalized $\tau$-moves, yielding a map $\hat{\phi}$ that reduces any word to a geodesic representative in a finite, predictable number of steps. The authors prove that two geodesic words representing the same element can be connected by length-preserving $\tau$-moves and commutations, establishing a form of length-preserving equivalence among geodesics and showing that $3$-free Artin groups satisfy Dehornoy's Property $H$. They also explain why the $3$-free condition is crucial: allowing length-$3$ relations breaks the canonical RRS framework and undermines the algorithm. Overall, the work provides an explicit, efficient algorithm for the word problem in a broad family of Artin groups and contributes to the understanding of their algorithmic structure and weak hyperbolicity properties.

Abstract

We give a quadratic-time explicit and computable algorithm to solve the word problem for Artin groups that do not contain any relations of length 3. Furthermore, we prove that, given two geodesic words representing the same element, one can obtain one from the other by using a set of homogeneous relations that never increase the word length.

Figures (2)

• Figure 1: Diagram for an RRS $U=u_1,u_2,u_3$ with $w_3=\epsilon$. (Notice by Lemma \ref{['lemma:beta']} that $\beta_2=\epsilon$).
• Figure 2: Diagram for an RRS $U=u_1,u_2$ with $u_2=aw_2$. (Notice by Lemma \ref{['lemma:beta']} that $\beta_1=\epsilon$).

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4: Property H
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Proposition 2.5