Dual structures on Coxeter and Artin groups of rank three

Delucchi, Emanuele¹ ; Paolini, Giovanni² ; Salvetti, Mario³

¹ IDSIA, University of Applied Arts and Sciences of Southern Switzerland, Lugano, Switzerland
² Dipartimento di Matematica, Università di Bologna, Bologna, Italy
³ Dipartimento di Matematica, Università di Pisa, Pisa, Italy

Geometry & topology, Tome 28 (2024) no. 9, pp. 4295-4336.

Résumé

We extend the theory of dual Coxeter and Artin groups to all rank-three Coxeter systems, beyond the previously studied spherical and affine cases. Using geometric, combinatorial, and topological techniques, we show that rank-three noncrossing partition posets are EL–shellable lattices and give rise to Garside groups isomorphic to the associated standard Artin groups. Within this framework, we prove the $K(\pi ,1)$ conjecture, the triviality of the center, and the solubility of the word problem for rank-three Artin groups. Some of our constructions apply to general Artin groups; we hope they will help develop complete solutions to the $K(\pi ,1)$ conjecture and other open problems in the area.

Détail

DOI : 10.2140/gt.2024.28.4295

Keywords: Artin groups, Coxeter groups, Garside structures, discrete Morse theory, shellability

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