On the structure of conjugation-free fundamental groups of conic-line arrangements

Michael Friedman, David Garber*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkable properties: either it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we first give a complete proof to the following key lemma: if we draw a new line through only one intersection point of a given real line arrangement whose fundamental group is conjugation-free, then the fundamental group of the new arrangement is also conjugation-free. Second, we generalize this lemma to the case of conic-line arrangements. Moreover, we prove that once the graph associated to conic-line arrangements (defined slightly different than the corresponding graph for line arrangements) has no cycles, then the fundamental group of its complement has a conjugation-free geometric presentation and in addition can be written as a direct sum of free groups and a free abelian group. Also, we show that if the graph consists of one cycle, and the conic does not pass through all the multiple points corresponding to the vertices of the cycle, then the fundamental group has a conjugation-free geometric presentation as well. For conclusion, we extend the family of real line arrangements having a conjugation-free geometric presentation (for their fundamental group) by defining the notion of a conjugation-free graph. We also extend this notion to certain families of conic-line arrangements.

Original languageEnglish
Pages (from-to)685-734
Number of pages50
JournalJournal of Homotopy and Related Structures
Volume10
Issue number4
DOIs
StatePublished - 1 Dec 2015
Externally publishedYes

Keywords

  • Conic-line arrangement
  • Conjugation-free graph
  • Conjugation-free presentation
  • Fundamental group
  • Hyperplane arrangement

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