Conjugation-free geometric presentations of fundamental groups of arrangements

2008/10/31 by Meital Eliyahu, Eliyahu, Meital, David Garber +3
Engineering · Mathematics · #14H30 (Primary) #32S22 #57M05 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Mathematics and Applications #graph theory and CDMA systems #math.AT #math.GR #math.GT #msc:14H30 #msc:32S22 #msc:57M05

paper · pdf · doi:10.48550/arxiv.0810.5615

28 pages, many figures; totally revised version; submitted

openalex publication_date 2008/10/31 · arxiv created 2010/03/16 · arxiv updated 2010/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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