The center of thin Gaussian groups
type: Item
instance of: scholarly article
title: The center of thin Gaussian groups
zbMATH Open document ID: 1002.20022
author: Matthieu Picantin
published in: Journal of Algebra
publication date: 2002-10-16T00:00:00Z
reviewed by: Carl Droms
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 1721739
zbMATH Keywords: thin Gaussian groups
zbMATH Keywords: Artin groups
zbMATH Keywords: quasi-centers
zbMATH Keywords: cancellative monoids
zbMATH Keywords: groups of fractions
MaRDI profile type: Publication
OpenAlex ID: W2046657616
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DOI: 10.1006/JABR.2001.8894
review text: A `thin Gaussian monoid' is a cancellative monoid \(M\) in which \(1\) is the only invertible element, in which least common multiples exist, and which admits a finite generating set which is closed under the operation \(\setminus\), where for any \(a,b\in M\), \(a(a\setminus b)\) is the right lcm of \(a\) and \(b\). A `thin Gaussian group' is the group of fractions of a thin Gaussian monoid. Thin Gaussian groups are a generalization of Artin groups of finite type (e.g., the braid groups).NEWLINENEWLINENEWLINEThe author analyzes the centers of thin Gaussian groups. He identifies a certain subclass (the `\(\Delta\)-pure' thin Gaussian groups) whose centers are infinite cyclic, generated by a power of an element \(\Delta\) which is analagous to the fundamental element of a finite-type Artin group. He then shows that every thin Gaussian group is a certain kind of product of \(\Delta\)-pure thin Gaussian groups.
description: scientific article; zbMATH DE number 1721739