On the monoid in the fundamental group of type $\mathrm{B_{ii}}$

Abstract

We study the monoid generated by certain Zariski-van Kampen generators in the positive homogeneous presented fundamental group of the complement of the logarithmic free divisor, called the type $\mathrm{B_{ii}}$ in the list by Sekiguchi. Although the monoid is cancellative, it turns out that the monoid is not Gaussian and, hence, is neither Garside nor Artin. Nevertheless, we show that the monoid carries certain particular elements similar to the fundamental elements in Artin monoid. Hence, we can solve the word problem and the conjugacy problem in the monoid and determine the center of it and the explicit form of the growth function for it. As a result, we can also solve the word problem and the conjugacy problem in the fundamental group, and determine the center of it (Theorem 5.8).