Advances in Pure Mathematics, 2011, 1, 218-220
doi:10.4236/apm.2011.14038 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Axes of Möbius Transformations in H3*
Chang-Jun Li, Li-Jie Sun, Na Li
School of Mathematical Sciences, Ocean University of China, Qingdao, China
Received March 11, 2011; revised March 28, 2011; accepted April 10, 2011
Abstract
This paper gives the relationship between the positions of axes of the two nonparabolic elements that gener-
ate a discrete group and the nature including the translation lengths along the axes and the rotation angles.
We mainly research the intersecting position and the coplanar (but disjoint) position.
Keywords: Geodesic, Discrete, Axis
1. Introduction
Hyperbolic 3-space is the set


33
123 3
=,, :>0HxxxRx
endowed with the complete Riemannian metric
d=
s
3
d
x
x of constant curvature equal to –1. A
Kleinian group G is a discrete nonelementary subgroup
of )
3
(
I
som H
, where 3
()
I
som H
 is the group of orien-
tation preserving isometries.
Each Möbius transformation of 3 =CH extends
uniquely via the Poincare’ extension [1] to an orien-
tation-preserving isometry of hyperbolic 3-space 3
H
. In
this way we identify Kleinian groups with discrete
Möbius groups.
Let M denote the group of all Möbius transformations
of the extended complex plane

=CC . We
associate with each Möbius transformation
=,
az b
fMadb
cz d

=1c
2,)
the matrix
=(
ab SL C
cd




A
And set , where denotes
the trace of the matrix
 
=trftr A

=tra dA
A
. Next, for each
f
and
g
in
M
we let [,]
f
g the multiplicatie commutator
11
denotev
f
gf g

. Wree complex numbers e call the th


11
,= 2f gtrfgfg 





22
=4,=ftrf gtrg  4


the parameters of ,
f
g
ice of
. These parameters are inde-
pendent of the cho matrix representation for s
f
and
g
in (2, )SL Cd they determine , an,
f
g uni-
quely up to conjugacy whenever

,0fg  .
Thelements of e
f
of
M
, other than thentity,
fall into three types.
e id
1) Elliptic:


[4,0f   and )
f
is conjugate to
zz

 where =  1
ic:
.
2) Loxodrom

 

4,0 d f  an
f
is conju-
zz
gate to

 where=1  ;
f
is hyperbolic if, in
addition, >0

.
3) Pa rabolic:


=0f  nd a
f
is conjugate to
zz

.
If
f
M

is nonp
f
arabolic, then fixes two points
C and th
points i
of e closed hyperbolic line joining these two
fixed s called the axis of
f
, noted by de


ax f.
In this case,
f
translates along


ax f by an amount


0f  , the translation lengt of h
f
,
f
about
rotates


ax f by an angle

]f  , and

(,
 
2
=4sin2
f
if
f






In [4], F.W.Gehring and G. J. Martin havehown :
Theorem 1.1: [4] If
s
,
f
g is discrete, if and f
g
are loxodromics with
 
=
f
g  , and if


f ax
and


ax g intersect at an angle

where 0< <


,
n the



 
sinhsinf

 
*The Project-sponsored by SRF for ROCS, SEM and NSFC (No.1077
1200). where . In particular,
0.122 0.435




C.-J. LI ET AL.
219


f

 
where 0.122 0.492

 . The exponent


sin

of
cannot nstant greater than 1.
In this paper, we will discuss the situation when
be replaced by a co


ax f and


ax g copla F. W.
situation
w
the following, we w
emma 2.1: [1] Let
nar but disjoint. In [4],
Gehring and G. J. Martin have analyzed the
hen f is loxodromic and g is loxodromic or elliptic. In
ill consider the condition when the
two generators are elliptics.
2. Preliminary Results
L
f
and
g
hen
be Möbius transforma-
tions, neither the identity. T
f
and
g
are conju-
gate if and only if

2
=tr ft

r g

2
.
Lemma 2.2: [4] If ,
f
g is a Kleinian group, if
f
is elliptic of order 3n, and if
g
is no of order 2,
then
t

,
f
gan  
where





2cos271if= 3
2cos25if= 4,5
=2cos26if= 6
2cos 21if7
n
n
an n
nn





 

Lemma 2.3: [3] Suppose that
f
and
g
in
M
have disjoint pairs of fixed points in C and

is
hyperbolic line in
the
3
H
which is oron
of
thogal to the axes
f
and
g
. Then

 
2
4, =sinh fg i
fg





where


 
=,= ,

f
gaxisfaxisg  
between the sphere or hyperplanes
and φ is the
angle hich contain w

ax f

 and

ax g

 respectively.
ma 2.4: [4] For each loxodrLem omic Möbius
transformation f there exists an integer 1m such
that

 
4sinh
3
m
f
f



The coefficient of



sinh
f
 cannot be replaced by
smaller constant.
3. Main Results
heorem 3.1: If T,
f
g is discrete, if
f
and
g
are
, n respectively e m,≥ 3,
en
elliptics with orders mwher n
th
1) If


ax f and


g intersect at an angle ax

where 0< <


, then
 
3
sin sina nm
   sin 2

2) If


ax f and


ax g are coplanar but disjoint,
then
 
3
sin sina nm
   sin 2
and the inequality is sharp.
Proof. Let

denote the hyperbolic distance between


ax f and


ax g. Let φ denote the the angle between
which containthe sphere or hyperplanes


ax f


and


ax g

 respectively. If

is the hyperbolic
3
line in
H
torthogonal to

ax f and hat is


ax g,
then


 
2
4, =sinh
fg

i
fg

 
by Lemma 2.3. If



ax f and intersect at an
angle

ax g

, then


 
2
4, si 
n
fg
fg


We may assume without loss of generality that

f
,
g
n are primitive elliptics. From Lemma 2.2 we can obtai




,3fga  , so

 
 

2
sin 4 ,
43
222
16sinsin sinmn
f
gfg
a
  


that is
 
 
3
sin sinsin2
a
nm   
In the same way, if ax(f) and ax(g) are coplanar but
disjoint, then

 

222
16sinsin sinmn
 

2
sinh 4 ,
43
f
gfg
a
  

by






 
2
4, sinh fg
fg



 To show that the inequa-
lity is sharp, we let ,
f
g denote the (2,3,7) triangle
group where
f
and
g
are primitive with

C.-J. LI ET AL.
220
37
f==g

2=
f
gI. Then

 
 


22
22
,= ]2=4
2
=4=2 cos2
fgtrtrf trg
fg
 [,
2
cos
73
=4=3
fg
trftr ga




 




 


Remark: In [4], according to Lemma 2.3, F. W.
Gehring and G. J. Martin considered the situation when
=0

. They discuss the relationship between the angle

, translation length of f and
g
or rotation angle when
f
is loxodromic and
g
is loxodromic or elliptic.
Theorem3.1 show the condition when
f
and
g
are
elliptics.
Corollary 3.1: If ,
f
g is discrete, if
f
and
g
e elliptics with

ar


=
f
g  ,

,0fg   and if

ax fand

ax g intersect at an angle

, where
0< 

2. If the order of
f
is with , then k3k
 
2sin 2
a
k


 


3
sin
In part eet
angles and the order
icular, if

ax fand

x g mat righta
of
f
is k, then
36k
Proof.
 
23
sin
sin
a
k



 asily seen from
2
  can e
the former theorem. If


ax f and

ax g meet at
right angles, then
2(3) = 0.248
sin 2
a
k





As k is an integer, so
36k

In the following, we will consider th thing wheen
=0

.
Theorem 3.2: If ,
f
g is discrete, if and
are loxodromics with
fg

=

f
g 


ax f and if

f and

ax g coplanar but disjoint, let ax


f
 be
nslation length of f, the tra

be the distance between
the


ax f and


ax gn
  
, the

3
sinhsinh 2 f  
whe
d
re =21cos7 d




 
.




By Lemma 2.4, can choose an integer
such that
Proof. we
number 1m

 
4sinh
m
3
f
f 


,
mm
f
gThen is a discrete noneleentary group
with
m




=
mm
fg  .
mma 2.2 aBy Lend Lemma 2.3, we can obtain
  



2
4sinh sinh
sinh
=4 ,
2
mm
mm
fg
fg
d

f 
3





then
  
3
sinhsinh 2π
d
f  

As for Theorem 3.5 and Theorem 3.15 in [
obtain related results in similar way when
4], we can


ax f and


ax g coplanar but disjoint.
4. Acknowledgements
The authors want to express theirs t
ymous referee for his valu
hanks to the ano-
able suggestions and pro-
n, “The Geometry of Discrete Groups,”
New York, 1983, p. 66.
. 12, 1995, pp.
n
fessor Qi-Zhi Fang for her support.
5. References
[1] A. F. Beardo
Spring-Verlag,
[2] C. Cao, “Some Trace Inequalities for Discrete Groups of
Möbius Transformations,” Proceedings of the American
Mathematical Society, Vol. 123, No
3807-3815. doi:10.2307/2161910
[3] F. W. Gehring and G. J. Martin, “Commutators, Collars
and the Geometry of Möbius Groups,” Journal d’Analyse
Mathématique, Vol. 63, No. 1, 1994, pp. 175-219.
doi:10.1007/BF03008423
[4] F. W. Gehring and G. J. Martin, “Geodesics in Hyper-
bolic 3-Folds,” Michigan Mathematical Journal, Vo
No. 2, 1997, pp. 331-343. doi:10.1307/mmj/1029005708
l. 44,
[5] F. W. Gehring and G. J. Martin, “Inequalities for Möbius
Transformations and Discrete Groups,” Journal für die
Reine und Angewandte Mathematik, No. 418, 1991, pp.
31-76. doi:10.1515/crll.1991.418.31