Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the structure of conjugation-free fundamental groups of conic-line
  arrangements   The fundamental group of the complement of a hyperplane arrangement plays an
important role in studying the corresponding arrangements. In particular, for
large families of hyperplane arrangements, this fundamental group, being
isomorphic to the fundamental group of a complement of a line arrangement, has
some remarkable properties: either it is a direct sum of free groups and a free
abelian group, or it has a conjugation-free geometric presentation.
  In this paper, we first give a complete proof to the following key lemma: if
we draw a new line through only one intersection point of a given real line
arrangement whose fundamental group is conjugation-free, then the fundamental
group of the new arrangement is also conjugation-free.
  Second, we generalize this lemma to the case of conic-line arrangements.
Moreover, we prove that once the graph associated to conic-line arrangements
(defined slightly different than the corresponding graph for line arrangements)
has no cycles, then the fundamental group of its complement has a
conjugation-free geometric presentation and in addition can be written as a
direct sum of free groups and a free abelian group. Also, we show that if the
graph consists of one cycle, and the conic does not pass through all the
multiple points corresponding to the vertices of the cycle, then the
fundamental group has a conjugation-free geometric presentation as well.
  For conclusion, we extend the family of real line arrangements having a
conjugation-free geometric presentation (for their fundamental group) by
defining the notion of a conjugation-free graph. We also extend this notion to
certain families of conic-line arrangements.

14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an