Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Commensurability of hyperbolic manifolds with geodesic boundary   Suppose n>2, let M,M' be n-dimensional connected complete finite-volume
hyperbolic manifolds with non-empty geodesic boundary, and suppose that the
fundamental group of M is quasi-isometric to the fundamental group of M' (with
respect to the word metric). Also suppose that if n=3, then the boundaries of M
and of M' are compact. We show that M is commensurable with M'. Moreover, we
show that there exist homotopically equivalent hyperbolic 3-manifolds with
non-compact geodesic boundary which are not commensurable with each other.
  We also prove that if M is as above and G is a finitely generated group which
is quasi-isometric to the fundamental group of M, then there exists a
hyperbolic manifold with geodesic boundary M'' with the following properties:
M'' is commensurable with M, and G is a finite extension of a group which
contains the fundamental group of M'' as a finite-index subgroup.

19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci