Monoids in the fundamental groups of the complement of logarithmic free divisors in C^3

TL;DR

This paper investigates the algebraic structure of monoids generated by fundamental group elements of complements of logarithmic free divisors in C^3, classifying their types and properties, and introducing concepts analogous to Artin monoids.

Contribution

It classifies the monoids arising from these fundamental groups, identifies which are Artin or free abelian, and introduces the notion of fundamental elements for non-Gaussian monoids.

Findings

Five monoids are Artin monoids.

Eight monoids are free abelian monoids.

Four monoids are neither Gaussian nor Garside.

Abstract

We study monoids generated by Zariski-van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C^3 listed by Sekiguchi (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaussian and, hence, are neither Garside nor Artin (Theorem 2). However, we introduce, similarly to Artin monoids, fundamental elements and show their existence (Theorem 3). One of the four non-Gaussian monoids satisfies the cancellation condition (Theorem 4).