On the structure of fundamental groups of complements of conic-line arrangements

Details On the structure of fundamental groups of complements of conic-line arrangements On the structure of fundamental groups of complements of conic-line arrangements

2011-11-22

arxiv.org/abs/1111.5291v1

Mathematics

Geometric Topology

14 pages, 8 figures; submitted

Scientific paper

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying these arrangements. In particular, for large families of these arrangements, this fundamental group has some remarkable properties: either it is a sum of free groups and a free abelian group, or, more generally, it has a conjugation-free geometric presentation. In this paper, we generalize these ideas to the case of conic-line arrangements. Explicitly, 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 is a direct sum of free groups and a free abelian group.

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