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

2010/09/07 by Meital Eliyahu, Eliyahu, Meital, David Garber +3
Mathematics · Computer Science · #Geometric and Algebraic Topology #semigroups and automata theory #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1009.1349

Abstract

A conjugation-free geometric presentation of a fundamental group is a\npresentation with the natural topological generators x₁, ..., x_n and the\ncyclic relations: x_ikx_i_k-1 ... x_i1 = x_i_k-1 ... x_i1\nx_ik = ... = x_i1 x_ik ... x_i2 with no conjugations on the\ngenerators.\n We have already proved that if the graph of the arrangement is a disjoint\nunion of cycles, then its fundamental group has a conjugation-free geometric\npresentation. In this paper, we extend this property to arrangements whose\ngraphs are a disjoint union of cycle-tree graphs.\n Moreover, we study some properties of this type of presentations for a\nfundamental group of a line arrangement's complement. We show that these\npresentations satisfy a completeness property in the sense of Dehornoy, if the\ncorresponding graph of the arrangement has no edges. The completeness property\nis a powerful property which leads to many nice properties concerning the\npresentation (such as the left-cancellativity of the associated monoid and\nyields some simple criterion for the solvability of the word problem in the\ngroup).\n

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