[Paper Review] Multifraction reduction IV: Padding and Artin-Tits groups of sufficiently large type

Patrick Dehornoy, Derek F. Holt+1 more|arXiv (Cornell University)|Jan 23, 2017

Geometric and Algebraic Topology7 references4 citations

TL;DR

This paper introduces padded multifraction reduction as a tool to weaken the convergence requirement for solving the word problem in Artin–Tits groups. It proves that for Artin–Tits monoids of sufficiently large type—where no triangle in the Coxeter diagram has exactly one edge labeled 2—$Σ$-reduction is semi-convergent up to computable padding, thereby establishing decidability of the word problem for this class via the reduction approach.

ABSTRACT

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.

Motivation & Objective

• To extend the applicability of multifraction reduction to Artin–Tits groups beyond the known FC type by introducing a weakened convergence condition.
• To establish that padded semi-convergence of $Σ$-reduction is sufficient for decidability of the word problem in the enveloping group.
• To connect padded semi-convergence with Property H and split reduction ($Σ$-reduction) in gcd-monoids.
• To prove that Artin–Tits monoids of sufficiently large type satisfy the padded version of Conjecture A, thus extending the scope of the reduction method.
• To provide a quadratic upper bound on the required padding for such groups, making the result effective.

Proposed method

• Introduce $f$-padding, defined as inserting an even number of trivial components at the start of a multifraction, to weaken the convergence condition.
• Define padded semi-convergence of $Σ$-reduction as a sufficient condition for decidability of the word problem in the enveloping group.
• Establish equivalence between padded semi-convergence of $Σ$-reduction, semi-convergence of split reduction ($Σ$-reduction), and Property H for lcm-presentations of gcd-monoids.
• Use a detailed analysis of the techniques from [10] and [8] to show that in sufficiently large type Artin–Tits monoids, the required padding is bounded by a quadratic function of the input word length.
• Apply the framework of subword reversing and fractional normal forms to transform multifractions into canonical representatives via $Σ$-reductions.
• Leverage the fact that Property H holds for sufficiently large type Artin–Tits monoids, which implies the required semi-convergence up to padding.

Experimental results

Research questions

• RQ1 Can the word problem for Artin–Tits groups of sufficiently large type be solved using multifraction reduction, despite the failure of full convergence?
• RQ2 Is there a computable padding function such that $Σ$-reduction becomes semi-convergent up to padding in sufficiently large type Artin–Tits monoids?
• RQ3 How does padded semi-convergence relate to Property H and split reduction ($Σ$-reduction) in gcd-monoids?
• RQ4 Can the reduction method be extended beyond FC-type Artin–Tits groups using this weakened convergence condition?
• RQ5 What is the quantitative bound on the padding required for semi-convergence in sufficiently large type Artin–Tits monoids?

Key findings

• The paper proves that for all Artin–Tits monoids of sufficiently large type, $Σ$-reduction is semi-convergent up to a computable padding function, satisfying the weakened Conjecture A${}^{ ext{padded}}$.
• A quadratic upper bound on the required padding is established, specifically $N \geq 4\sum_{i=1}^{k}|v_{i}|$, where $k$ is the number of subwords and $|v_i|$ their lengths.
• Padded semi-convergence of $Σ$-reduction is equivalent to Property H holding for the lcm-presentation of the monoid.
• The result confirms that the multifraction reduction approach is relevant for a significantly larger class of Artin–Tits groups than previously known, including those of sufficiently large type.
• The word problem for Artin–Tits groups of sufficiently large type is decidable via the reduction method, provided the padding is computable.
• The paper leaves open the case of the monoid with exponents $3,3,3,3,3,2$, which remains the first unresolved case for the reduction method.