A CONJUGATION-FREE GEOMETRIC PRESENTATION OF FUNDAMENTAL GROUPS OF ARRANGEMENTS II: EXPANSION AND SOME PROPERTIES

MEITAL ELIYAHU, DAVID GARBER, MINA TEICHER
2011 International journal of algebra and computation  
A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x 1 , . . . , x n and the cyclic relations: with no conjugations on the generators. We have already proved in [13] that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs.
more » ... ee graphs. Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement is triangle-free. The completeness property is a powerful property which leads to many nice properties concerning the presentation (as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).
doi:10.1142/s0218196711006479 fatcat:476vbaabszerzawbihnpbw4oma