Hiroshima Mathematical Journal

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

Tadashi Ishibe

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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).

Article information

Source
Hiroshima Math. J., Volume 42, Number 1 (2012), 99-114.

Dates
First available in Project Euclid: 30 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1333113008

Digital Object Identifier
doi:10.32917/hmj/1333113008

Mathematical Reviews number (MathSciNet)
MR2952074

Zentralblatt MATH identifier
1261.20065

Subjects
Primary: 20F05: Generators, relations, and presentations

Keywords
Monoid fundamental group the word problem the conjugacy problem

Citation

Ishibe, Tadashi. On the monoid in the fundamental group of type $\mathrm{B_{ii}}$. Hiroshima Math. J. 42 (2012), no. 1, 99--114. doi:10.32917/hmj/1333113008. https://projecteuclid.org/euclid.hmj/1333113008


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