There are three scale factors so that after radial projection on the unit hyperboloid, we get a billiard in two-dimensional hyperbolic space. The billiard region is defined by the following relevant wall inequalities,
(symmetry walls) and (curvature wall). The remarkable properties of this region from our point of view are:These angles are captured in the matrix of scalar products,
Because the angles between the reflecting planes are integer submultiples of , the reflections in the walls bounding the
billiard region6,
The group generated by the reflections ,
and
is denoted
, for reasons that will
become clear in the following, and coincides with the arithmetic group
, as we will now show
(see also [75
, 116
, 107
]).
The group is defined as the group of
matrices
with integer entries and
determinant equal to
, with the identification of
and
,
There are two interesting realisations of in terms of transformations in two
dimensions:
The transformation (3.14) is the composition of the identity with the transformation (3.11
)
when
, and of the complex conjugation transformation,
with the
transformation (3.11
) when
. Because the coefficients
,
,
, and
are real,
commutes with
and furthermore the map (3.11
)
(3.14
) is a group isomorphism, so that
we can indeed either view the group
as the group of fractional transformations (3.11
), or
as the group of transformations (3.14
).
An important subgroup of the group is the group
for which
, also
called the “modular group”. The translation
and the inversion
are
examples of modular transformations,
Let ,
and
be the
-transformations
One easily verifies that and that
. Since any transformation of
not in
can be written as a transformation of
times, say,
and
since any transformation of
can be written as a product of
’s and
’s, it follows that the
group generated by the 3 reflections
,
and
coincides with
, as announced
above. (Strictly speaking,
could be a quotient of that group by some invariant
subgroup, but one may verify that the kernel of the homomorphism is trivial (see Section 3.2.5
below).) The fundamental domains for
and
are drawn in Figure 2
. The
equivalence between
and the Coxeter group
has been discussed previously
in [75
, 116
, 107
].
|
http://www.livingreviews.org/lrr-2008-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |