TY - JOUR
T1 - A conjugation-free geometric presentation of fundamental groups of arrangements II
T2 - Expansion and some properties
AU - Eliyahu, Meital
AU - Garber, David
AU - Teicher, Mina
N1 - Funding Information: ‖Partially supported by the Israeli Ministry of Science and Technology.
PY - 2011/8
Y1 - 2011/8
N2 - A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x1,..,x n and the cyclic relations: $$ x-{i-k}x-{i-{k-1}} \cdots xi1} = xik-1} \cdots xi1} xik} = \cdots = xi1} xik} \cdots xi2} $$ 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. 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 (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).
AB - A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x1,..,x n and the cyclic relations: $$ x-{i-k}x-{i-{k-1}} \cdots xi1} = xik-1} \cdots xi1} xik} = \cdots = xi1} xik} \cdots xi2} $$ 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. 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 (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).
KW - Conjugation-free presentation
KW - complemented presentation
KW - complete presentation
KW - fundamental group
UR - http://www.scopus.com/inward/record.url?scp=80051720355&partnerID=8YFLogxK
U2 - https://doi.org/10.1142/S0218196711006479
DO - https://doi.org/10.1142/S0218196711006479
M3 - مقالة
SN - 0218-1967
VL - 21
SP - 775
EP - 792
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
IS - 5
ER -