Multifraction reduction II: Conjectures for Artin–Tits groups

Abstract

Multifraction reduction is a new approach to the word problem for Artin-Tits groups and, more generally, for the enveloping group of a monoid in which any two elements admit a greatest common divisor. This approach is based on a rewrite system ("reduction") that extends free group reduction. In this paper, we show that assuming that reduction satisfies a weak form of convergence called semi-convergence is sufficient for solving the word problem for the enveloping group, and we connect semi-convergence with other conditions involving reduction. We conjecture that these properties are valid for all Artin-Tits monoids, and provide partial results and numerical evidence supporting such conjectures.Conjecture A. Reduction is semi-convergent for every Artin-Tits monoid.Proof. The argument is similar to the proof of [13, Proposition 2.4], and we only point the differences due to using signed multifractions. For (i), associativity is checked directly, and the generating subsets for F ± M and F M follow from the equalities (2.7) a 1 /···/a n = a 1 · a 2 · a 3 · a 4 · ··· = a 1 · 1/a 2 · a 3 · 1/a 4 · ··· :both hold in F ± M , and the second only involves positive multifractions. For (ii) and (iii), for every a in F ± M , the definition of ≃ ± implies ι(a) = ι(a) −1 and, writing ι + (a) for the ≃-class of a, that of ≃ implies ι + (1/a) = ι + (a) −1 . Hence both F ± M /≃ ± and F M /≃ are groups generated by M . One easily checks that the latter groups satisfy the universal property defining U(M ), and are therefore isomorphic to U(M ). Then (2.6) directly follows from (2.7).Next, for every a in F ± M , the product 1 · a belongs to F M . Then, for a, b in F ± M , write a ≈ b for 1 · a ≃ 1 · b. By considering all sign combinations and using relations like 1/ab ≃ 1/b · 1/a, However, a number of Artin-Tits monoids fail to be of type FC and therefore are not eligible for Proposition 2.19, typically the monoid of type A 2 considered in Example 2.16. So we are left with the question of either weakening the assumptions for Proposition 2.19, or using a conclusion weaker than convergence. Semi-convergenceAfter showing in Subsection 3.1 that the 3-Ore assumption cannot be weakened when proving the convergence of R ± M , we introduce in Subsection 3.2 a new property of R ± M called semiconvergence, which, as the name suggests, is weaker than convergence. We conjecture that, for every Artin-Tits monoid, the system R ± M is semi-convergent ("Conjecture A"). We prove in Subsection 3.3 that most of the consequences known to follow from the convergence of R ± M follow from its semi-convergence, in particular in terms of controlling the group U(M ) from inside the monoid M and solving its word problem. Finally, we describe in Subsection 3.4 several variants of semi-convergence.3.1. The strength of the 3-Ore condition. A first attempt for improving Proposition 2.19 could be to establish the convergence of R ± M from an assumption weaker than the 3-Ore condition. This approach fails, as the latter turns out to be not only sufficient, but also ...

Multifraction reduction IV: Padding and Artin–Tits monoids of sufficiently large type

“…Proof of Proposition 1.6. The argument is the same as the one in [3] for Proposition 1.4. Let S be the atom set of M , and let w be a word in S∪S.…”

“…By [2, Lemma 3.8 and Cor. 3.20], a ⇒ * b implies that a and b represent the same element in U(M ), and, conversely, the relation ι(a) = ι(b) is essentially the equivalence relation generated by ⇒ (up to deleting trivial final entries). Furthermore, whenever the monoid M is noetherian, R-reduction is terminating for M .…”

“…In the definition of multifraction reduction used in [2,3,6] and repeated in Section 1.2, in order for a • R i,x to be defined, x and a i have to admit a common multiple, a left-or right-multiple depending on the parity of i. In order to define a variant of this reduction, we now relax the assumption and only require that x and some divisor of a i admit a common multiple; then, we again remove x from a i+1 and push it to the left or right side using the lcm operation, this time through the divisor of a i that is involved and not necessarily through the whole of a i .…”

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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 H. As an application, we show that all Artin-Tits groups of sufficiently large type satisfy some weakening Conjecture A padded of Conjecture A, thus showing that the reduction approach is relevant for these groups.Reduction of multifractions, which was introduced in [2] and [3], is a new approach to the word problem for Artin-Tits groups and, more generally, for groups that are enveloping groups of monoids in which the divisibility relations have weak lattice properties ("gcd-monoids"). It is based on a rewrite system ("R-reduction") that extends the usual free reduction for free groups, as well as the rewrite systems known for Artin-Tits groups of spherical type, and more generally Garside groups.

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“…Reduction of multifractions, which was introduced in [2] and [3], is a new approach to the word problem for Artin-Tits groups and, more generally, for groups that are enveloping groups of monoids in which the divisibility relations have weak lattice properties ("gcd-monoids"). It is based on a rewrite system ("R-reduction") that extends the usual free reduction for free groups, as well as the rewrite systems known for Artin-Tits groups of spherical type, and more generally Garside groups.…”