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Cancellative Elliptic Artin Monoids

Georges Neaime

TL;DR

The paper addresses the lack of cancellativity in prior elliptic Artin monoid presentations, which blocked Garside-theoretic approaches. It constructs new, cancellative presentations for simply-laced elliptic Artin groups by introducing infinite families of generators $\{t_i\}_{i\in\mathbb{Z}}$ alongside the $s_j$, and proves isomorphisms to the standard elliptic groups while extending to types $\tilde{D}_4^{(1,1)}$ and $\tilde{E}_n^{(1,1)}$ for $n=6,7,8$. Using word reversing and cube-conditions, the authors show the new monoids $M'(\tilde{D}_4^{(1,1)})$ and $M'(\tilde{E}_n^{(1,1)})$ are cancellative, hence complete and complemented, which by Dehornoy's results yields cancellativity. This establishes a foundational step toward Garside structures for elliptic Artin monoids and groups, with potential implications for word problems and related conjectures in elliptic settings.

Abstract

We define new presentations for elliptic Artin groups. We also show that the elliptic monoids defined by these presentations are cancellative. This solves the failure of cancellativity for the presentations of elliptic Artin monoids that were introduced before in the literature. Our approach also paves the way to construct Garside structures for elliptic Artin monoids and groups.

Cancellative Elliptic Artin Monoids

TL;DR

The paper addresses the lack of cancellativity in prior elliptic Artin monoid presentations, which blocked Garside-theoretic approaches. It constructs new, cancellative presentations for simply-laced elliptic Artin groups by introducing infinite families of generators alongside the , and proves isomorphisms to the standard elliptic groups while extending to types and for . Using word reversing and cube-conditions, the authors show the new monoids and are cancellative, hence complete and complemented, which by Dehornoy's results yields cancellativity. This establishes a foundational step toward Garside structures for elliptic Artin monoids and groups, with potential implications for word problems and related conjectures in elliptic settings.

Abstract

We define new presentations for elliptic Artin groups. We also show that the elliptic monoids defined by these presentations are cancellative. This solves the failure of cancellativity for the presentations of elliptic Artin monoids that were introduced before in the literature. Our approach also paves the way to construct Garside structures for elliptic Artin monoids and groups.

Paper Structure

This paper contains 4 sections, 11 theorems, 6 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The affine Artin group of type $\tilde{A}$
  3. Presentations of elliptic Artin groups
  4. Cancellativity of elliptic Artin monoids

Key Result

Proposition 3.2

The elliptic Artin group $\textsc{Art}{(\tilde{D}_4^{(1,1)})}$ of type $\tilde{D}_4^{(1,1)}$ is isomorphic to the group $G'(\tilde{D}_4^{(1,1)})$ defined by a presentation with generators $t_i$ for $i\in \mathbb{Z}$ ,$s_1,s_2,s_3,s_4$ and relations: Adding the quadratic relations $t_i^2=1$ ($i \in \mathbb{Z}$) and $s_j^2=1$ for $j=1,2,3,4$, we obtain a presentation of the group $\textsc{Hyp}{(\ti

Figures (8)

  • Figure 1: The diagram of the presentation of $\textsc{Art}(\tilde{A}_{n-1})$ given in Definition \ref{['Def_StandardPres_TildeA']}.
  • Figure 2: The diagram of the presentation of $\textsc{Art}(\tilde{A}_{n-1})$ given in Definition \ref{['Def_PresShi_TildeA']}.
  • Figure 3: The diagram of the presentation of $\textsc{Art}(\tilde{A}_{n-1})$ given in Definition \ref{['Def_PresCorranLeeLee_TildeA']}.
  • Figure 4: A portion of the Coxeter complex of type $\tilde{A}_2$, where we show the $3$ generating sets of $\textsc{Cox}(\tilde{A}_{2})$ given in Definitions \ref{['Def_StandardPres_TildeA']}, \ref{['Def_PresShi_TildeA']}, and \ref{['Def_PresCorranLeeLee_TildeA']}.
  • Figure 5: Elliptic Dynkin diagram of type $\tilde{D}_4^{(1,1)}$.
  • ...and 3 more figures

Theorems & Definitions (32)

  • Definition 2.1: Classical presentation
  • Definition 2.2: Presentation of Shi
  • Definition 2.3: Presentation of Corran--Lee--Lee
  • Example 2.4
  • Definition 3.1: Presentation for type $\tilde{D}_4^{(1,1)}$
  • Proposition 3.2: New presentation for type $\tilde{D}_4^{(1,1)}$
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 22 more