Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Canonical Hexagons and the PSL(2,C) Discreteness Problem   The discreteness problem, that is, the problem of determining whether or not
a given finitely generated group G of orientation preserving isometries of
hyperbolic three-space is discrete as a subgroup of the whole isometry group of
hyperbolic three space, is a challenging problem that has been investigated for
more than a century and is still open. It is known that G is discrete if, and
only if, every non-elementary two generator subgroup is. Several sufficient
conditions for discreteness are also known as are some necessary conditions,
though no single necessary and sufficient condition is known. There is a finite
discreteness algorithm for the two generator subgroups of the isometry group of
hyperbolic two-space. But the situation in three dimensions is more delicate
because there are geometrically infinite groups.
  We present a semi-algorithm, that is, a procedure that terminates sometimes
but not always. There is no standard way to find an infinite sequence of
distinct elements that converges to the identity to show that a group is not
discrete. Our semi-algorithm either produces such an infinite sequence or finds
a finite sequence that produces a right angled hexagon in hyperbolic
three-space which has a special property that is a generalization of the notion
of convexity. We call it a canonical hexagon. If the group is discrete, free
and geometrically finite, it always has an essentially unique canonical hexagon
which the procedure finds in a finite number of steps.

31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph