![Figure 7: Intersections of the isometric spheres I 0 , I 1 , IC 1 and IC 1 with IC0 in the boundary of H 2 C , viewed in geographical coordinates. Recall that rC 0 D IC 0 \ I 0 and r 0 D IC 0 \ I 1 . Here ˛ 2 2 ; 2 is the vertical coordinate, and ˇ 2 Œ ; the horizontal one. The vertical dash-dotted segments ˇ D ˙ 2 are the two halves of the boundary of the meridian m . The bigon between the two curves rC 0 and r 0 is BC 0 (see Proposition 6.5). Compare to [8, Figure 2].](/figures/figure-7-intersections-of-the-isometric-spheres-i-0-i-1-ic-1-38xc932e.webp)
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Parker, John R. and Will, Pierre (2017) 'A complex hyperbolic Riley slice.', Geometry and topology., 21 (6).
pp. 3391-3451.
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Geometry & Topology 21 (2017) 3391–3451
A complex hyperbolic Riley slice
JOHN R PARKER
PIERRE WILL
We study subgroups of
PU.2; 1/
generated by two noncommuting unipotent maps
A
and
B
whose product
AB
is also unipotent. We call
U
the set of conjugacy classes of
such groups. We provide a set of coordinates on
U
that make it homeomorphic to
R
2
.
By considering the action on complex hyperbolic space
H
2
C
of groups in
U
, we
describe a two-dimensional disc
Z
in
U
that parametrises a family of discrete groups.
As a corollary, we give a proof of a conjecture of Schwartz for
.3; 3; 1/
–triangle
groups. We also consider a particular group on the boundary of the disc
Z
where
the commutator
ŒA; B
is also unipotent. We show that the boundary of the quotient
orbifold associated to the latter group gives a spherical CR uniformisation of the
Whitehead link complement.
20H10, 22E40, 51M10; 57M50
1 Introduction
1.1 Context and motivation
The framework of this article is the study of the deformations of a discrete subgroup
of a Lie group
H
in a Lie group
G
containing
H
. This question has been addressed in
many different contexts. A classical example is the one where
is a Fuchsian group,
H DPSL.2; R/
and
G DPSL.2; C/
. When
is discrete, such deformations are called
quasi-Fuchsian. We will be interested in the case where
is a discrete subgroup of
H D SO.2; 1/
and
G
is the group
SU.2; 1/
(or their natural projectivisations over
R
and
C
, respectively). The geometrical motivation is very similar: In the classical case
mentioned above,
PSL.2; C/
is the orientation-preserving isometry group of hyperbolic
3
–space
H
3
and a Fuchsian group preserves a totally geodesic hyperbolic plane
H
2
in
H
3
. In our case
G DSU.2; 1/
is (a triple cover of) the holomorphic isometry group
of complex hyperbolic
2
–space
H
2
C
, and the subgroup
H D SO.2; 1/
preserves a
totally geodesic Lagrangian plane isometric to
H
2
. A discrete subgroup
of
SO.2; 1/
is called
R
–Fuchsian. A second example of this construction is where
G
is again
SU.2; 1/
but now
H DS.U.1/ U.1; 1//
. In this case
H
preserves a totally geodesic
complex line in
H
2
C
. A discrete subgroup of
H
is called
C
–Fuchsian. Deformations of
Published: 31 August 2017 DOI: 10.2140/gt.2017.21.3391
3392 John R Parker and Pierre Will
either
R
–Fuchsian or
C
–Fuchsian groups in
SU.2; 1/
are called complex hyperbolic
quasi-Fuchsian. See Parker and Platis [24] for a survey of this topic.
The title of this article refers to the so-called Riley slice of Schottky space (see [19; 1]).
Riley considered the space of conjugacy classes of subgroups of
PSL.2; C/
generated
by two noncommuting parabolic maps. This space may be identified with
C f0g
under the map that associates the parameter
2 C f0g
with the conjugacy class of
the group
, where
D
1 1
0 1
;
1 0
1
:
Riley was interested in the set of those parameters
for which
is discrete. He was
particularly interested in the (closed) set where
is discrete and free, which is now
called the Riley slice of Schottky space; see Keen and Series [19]. This work has been
taken up more recently by Akiyoshi, Sakuma, Wada and Yamashita. In their book [1]
they illustrate one of Riley’s original computer pictures,
1
Figure 0.2a, and their version
of this picture, Figure 0.2b. Riley’s main method was to construct the Ford domain
for
. The different combinatorial patterns that arise in this Ford domain correspond
to the differently coloured regions in these figures from [1]. Riley was also interested
in groups
that are discrete but not free. In particular, he showed that when
is
a complex sixth root of unity then the quotient of hyperbolic
3
–space by
is the
figure-eight knot complement.
1.2 Main definitions and discreteness result
The direct analogue of the Riley slice in complex hyperbolic plane would be the set
of conjugacy classes of groups generated by two noncommuting, unipotent parabolic
elements
A
and
B
of
SU.2; 1/
. (Note that in contrast to
PSL.2; C/
, there exist
parabolic elements in
SU.2; 1/
that are not unipotent. In fact, there is a
1
–parameter
family of parabolic conjugacy classes; see for instance Goldman [15, Chapter 6].) This
choice would give a four-dimensional parameter space, and we require additionally
that AB be unipotent, making the dimension drop to 2. Specifically, we define
(1) U D f.A; B/ 2 SU.2; 1/
2
W A; B; AB all unipotent and AB ¤ BAg=SU.2; 1/:
Following Riley, we are interested in the (closed) subset of
U
where the group
hA; Bi
is discrete and free and our main method for studying this set is to construct the Ford
domain for its action on complex hyperbolic space
H
2
C
. We shall also indicate various
other interesting discrete groups in U but these will not be our main focus.
1
Parker has one of Riley’s printouts of this picture dated 26th March 1979.
Geometry & Topology, Volume 21 (2017)
A complex hyperbolic Riley slice 3393
In Section 3.1, we will parametrise
U
so that it becomes the open square
2
;
2
2
.
The parameters we use will be the Cartan angular invariants
˛
1
and
˛
2
of the triples of
(parabolic) fixed points of
.A; AB; B/
and
.A; AB; BA/
, respectively (see Section 2.6
for the definitions). Note that the invariants
˛
1
and
˛
2
are defined to lie in the closed
interval
2
;
2
. Our assumption that
A
and
B
don’t commute implies that neither
˛
1
nor ˛
2
can equal ˙
2
(see Section 3.1).
When
˛
1
and
˛
2
are both zero, that is, at the origin of the square, the group
hA; Bi
is
R
–Fuchsian. The quotient of the Lagrangian plane preserved by
hA; Bi
is a hyperbolic
thrice-punctured sphere where the three (homotopy classes of) peripheral elements are
represented by (the conjugacy classes of)
A
,
B
and
AB
. The space
U
can thus be
thought of as the slice of the
SU.2; 1/
–representation variety of the thrice-punctured
sphere group defined by the conditions that the peripheral loops are mapped to unipotent
isometries.
We can now state our main discreteness result.
Theorem 1.1
Suppose that
DhA; Bi
is the group associated to parameters
.˛
1
; ˛
2
/
satisfying D.4 cos
2
˛
1
; 4 cos
2
˛
2
/ > 0, where D is the polynomial given by
D.x; y/ D x
3
y
3
9x
2
y
2
27xy
2
C81xy 27x 27:
Then is discrete and isomorphic to the free group F
2
. This region is Z in Figure 1.
Note that at the centre of the square, we have
D.4; 4/ D 1225
for the
R
–Fuchsian
representation. The region
Z
where
D > 0
consists of groups
whose Ford domain has
the simplest possible combinatorial structure. It is the analogue of the outermost region
in the two figures from Akiyoshi, Sakuma, Wada and Yamashita [1] mentioned above.
1.3 Decompositions and triangle groups
We will prove in Proposition 3.3 that all pairs
.A; B/
in
U
admit a (unique) decompo-
sition of the form
(2) A D ST and B D TS;
where
S
and
T
are order-three regular elliptic elements (see Section 2.2). In turn, the
group generated by
A
and
B
has index three in the one generated by
S
and
T
. When
either
˛
1
D 0
or
˛
2
D 0
there is a further decomposition making
hA; Bi
a subgroup
of a triangle group.
Deformations of triangle groups in
PU.2; 1/
have been considered in many places,
including Goldman and Parker [16], Parker, Wang and Xie [25], Pratoussevitch [28]
and Schwartz [32]. A complex hyperbolic
.p; q; r /
–triangle is one generated by three
complex involutions about (complex) lines with pairwise angles
p
,
q
, and
r
, where
Geometry & Topology, Volume 21 (2017)
3394 John R Parker and Pierre Will
4
5
6
7 7
P
2
1
0
3
Z
2
7
6
5
4
7
0 R–Fuchsian representation of the 3–punctured sphere group.
1
Horizontal segment corresponding to even word subgroups of ideal triangle groups; see
Goldman and Parker [16] and Schwartz [30; 31; 33].
2
Last ideal triangle group, contained with index three in a group uniformising the White-
head link complement obtained by Schwartz [30; 31; 33].
3
Vertical segment corresponding to bending groups that have been proved to be discrete
by Will [37].
4 .3; 3; 4/
–group uniformising the figure-eight knot complement. Obtained by Deraux
and Falbel [8].
5 .3; 3; n/
–groups, proved to be discrete by Parker, Wang and Xie [25]. On this picture,
4 6 n 6 8.
6 Uniformisation of the Whitehead link complement we obtain in this work.
7 Subgroup of the Eisenstein–Picard lattice; see Falbel and Parker [14].
Figure 1: The parameter space for
U
. The exterior curve
P
corresponds to
classes of groups for which
ŒA; B
is parabolic. The central dashed curve
bounds the region
Z
where we prove discreteness. The labels correspond
to various special values of the parameters. Points with the same labels are
obtained from one another by symmetries about the coordinate axes. The
results of Section 3.3 imply that they correspond to groups conjugate in
Isom(H
2
C
).
p
,
q
and
r
are integers or
1
(when one of them is
1
the corresponding angle
is
0
). Groups generated by complex reflections of higher order are also interesting;
see Mostow [22] for example, but we do not consider them here. For a given triple
.p; q; r /
with
minfp; q; r g > 3
, the deformation space of the
.p; q; r /
–triangle group
Geometry & Topology, Volume 21 (2017)