A conjugation-free geometric presentation of fundamental groups of arrangements

• DOI: 10.1007/s00229-010-0380-2
• Publication Date: 2010-07-11T21:13:42Z
• Authors: Meital Eliyahu; David Garber; Mina Teicher
• Keywords: Disjoint union (topology); Free product; Multiplicity (mathematics); Disjoint sets; Fundamental group; Complement

28 pages, many figures; totally revised version; submitted<br/>We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union of disjoint cycles, has no line with more than two multiple points, and where the multiplicities of the multiple points are arbitrary. We also compute the exact group structure (by means of a semi-direct product of groups) of the arrangement of 6 lines whose graph consists of a cycle of length 3, and all the multiple points have multiplicity 3.<br/>