Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the intersections of finitely generated subgroups of free groups:
  reduced rank to full rank   We show that the number of conjugacy classes of intersections $A\cap B^g$,
for fixed finitely generated subgroups $A, B<F$ of a free group, is bounded
above in terms of the ranks of $A$ and $B$; this confirms an intuition of
Walter Neumann. This result was previously known only in the case where $A$ or
$B$ is cyclic by the $w$-cycles theorem of Helfer and Wise and, independently,
Louder and Wilton. In order to prove our main theorem, we introduce new results
regarding the structure of fibre products of finite graphs. We work with a
generalised definition of graph immersions so that our results apply to the
theory of regular languages. For example, we give a new algorithm to decide
non-emptiness of the intersection of two regular languages.

15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con