A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties

Meital Eliyahu, David Garber, Mina Teicher

Research output: Contribution to journalArticlepeer-review

Abstract

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x1,..,x n and the cyclic relations: $$ x-{i-k}x-{i-{k-1}} \cdots xi1} = xik-1} \cdots xi1} xik} = \cdots = xi1} xik} \cdots xi2} $$ with no conjugations on the generators. We have already proved in [13] that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs. Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement is triangle-free. The completeness property is a powerful property which leads to many nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).

Original languageEnglish
Pages (from-to)775-792
Number of pages18
JournalInternational Journal of Algebra and Computation
Volume21
Issue number5
DOIs
StatePublished - Aug 2011

Keywords

  • Conjugation-free presentation
  • complemented presentation
  • complete presentation
  • fundamental group

All Science Journal Classification (ASJC) codes

  • General Mathematics

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