Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quasiperiodic and mixed commutator factorizations in free products of
  groups   It is well known that a nontrivial commutator in a free group is never a
proper power. We prove a theorem that generalizes this fact and has several
worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k,
y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups
without nontrivial elements of order $\le n$ implies that $z$ is conjugate to
an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a
free group factors into a product of elements which are conjugate to each other
then all these elements are distinct.

16
The categories of this paper are: stat
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
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
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin