 Advances in Pure Mathematics, 2011, 1, 218-220 doi:10.4236/apm.2011.14038 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. 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 E-mail: changjunli7921@hotmail.com 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 33123 3=,, :>0HxxxRx endowed with the complete Riemannian metric d=s3dxx of constant curvature equal to –1. A Kleinian group G is a discrete nonelementary subgroup of )3(Isom H, where 3()Isom H is the group of orien- tation preserving isometries. Each Möbius transformation of 3=CH extends uniquely via the Poincare’ extension  to an orien- tation-preserving isometry of hyperbolic 3-space 3H. 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 bfMadbcz d=1c2,) the matrix =(ab SL CcdA And set , where denotes the trace of the matrix  =trftr A=tra dAA. Next, for each f and g in M we let [,]fg the multiplicatie commutator 11 denotevfgf g. Wree complex numbers e call the th11,= 2f gtrfgfg 22=4,=ftrf gtrg4 the parameters of ,fgice of. These parameters are inde- pendent of the cho matrix representation for sf and g in (2, )SL Cd they determine , an,fg uni- quely up to conjugacy whenever ,0fg. Thelements of e f of M, other than thentity, fall into three types. e id1) Elliptic: [4,0f and )f is conjugate to zz where =1ic: . 2) Loxodrom4,0 d fanf is conju- zzgate to  where=1; f is hyperbolic if, in addition, >0. 3) Pa rabolic:=0fnd a f is conjugate to zz. If fM is nonpfarabolic, then fixes two points C and thpoints iof 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 hf, f about rotatesax f by an angle ]f, and (, 2=4sin2fiff In , F.W.Gehring and G. J. Martin havehown : Theorem 1.1:  If s ,fg is discrete, if and fg are loxodromics with  =fg, and if f axand ax g intersect at an angle  where 0< <, n thesinhsinf *The Project-sponsored by SRF for ROCS, SEM and NSFC (No.10771200). 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 wthe following, we wemma 2.1:  Let nar but disjoint. In , 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 Lf and g hen be Möbius transforma- tions, neither the identity. Tf and g are conju- gate if and only if 2=tr ftr g2. Lemma 2.2:  If ,fg is a Kleinian group, if f is elliptic of order 3n, and if g is no of order 2, then t,fgan where 2cos271if= 32cos25if= 4,5=2cos26if= 62cos 21if7nnan nnn  Lemma 2.3:  Suppose that f and g in M have disjoint pairs of fixed points in C and  is hyperbolic line in the 3H which is oronofthogal to the axes f and g. Then  24, =sinhfg ifg where =,= ,fgaxisfaxisg between the sphere or hyperplanes and φ is the angle hich contain wax f and ax g respectively. ma 2.4:  For each loxodrLem omic Möbius transformation f there exists an integer 1m such that 4sinh3mff The coefficient of sinhf cannot be replaced by smaller constant. 3. Main Results heorem 3.1: If T,fg is discrete, if f and g are , n respectively e m,≥ 3, en elliptics with orders mwher n th1) If ax f and g intersect at an angle ax where 0< <, then  3sin sinanm  sin 22) If ax f and ax g are coplanar but disjoint, then  3sin sinanm  sin 2and 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 3line in H torthogonal to ax f and hat is ax g, then  24, =sinhfgifg by Lemma 2.3. If ax f and intersect at an angle ax g, then  24, sinfgfg We may assume without loss of generality that f,g n are primitive elliptics. From Lemma 2.2 we can obtai,3fga, so   2sin 4 ,4322216sinsin sinmnfgfga   that is  3sin sinsin2anm  In the same way, if ax(f) and ax(g) are coplanar but disjoint, then  22216sinsin sinmn 2sinh 4 ,43fgfga  by  24, sinhfgfg To show that the inequa- lity is sharp, we let ,fg denote the (2,3,7) triangle group where f and g are primitive with Copyright © 2011 SciRes. APM C.-J. LI ET AL. Copyright © 2011 SciRes. APM 220 37f==g2=fgI. Then   2222,= ]2=42=4=2 cos2fgtrtrf trgfg[,2cos73=4=3fgtrftr ga   Remark: In , 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 ,fg is discrete, if f and g e elliptics with ar=fg, ,0fg and if ax fand ax g intersect at an angle , where 0<2. If the order of f is with , then k3k 2sin 2ak  3sinIn part eet angles and the ordericular, if ax fand x g mat righta of f is k, then 36k Proof.  23sinsinak asily seen from 2 can ethe former theorem. If ax f and ax g meet at right angles, then 2(3) = 0.248 sin 2ak As k is an integer, so 36k In the following, we will consider th thing wheen =0. Theorem 3.2: If ,fg is discrete, if and are loxodromics with fg =fg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3sinhsinh 2f whedre =21cos7d.  By Lemma 2.4, can choose an integer such that Proof. wenumber 1m4sinhm3ff ,mmfgThen is a discrete noneleentary group with m=mmfg. mma 2.2 aBy Lend Lemma 2.3, we can obtain 24sinh sinhsinh=4 ,2mmmmfgfgdf3 then 3sinhsinh 2πdf  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 tymous referee for his valuhanks to the ano- able suggestions and pro- n, “The Geometry of Discrete Groups,” New York, 1983, p. 66. . 12, 1995, pp. nfessor Qi-Zhi Fang for her support. 5. References  A. F. BeardoSpring-Verlag, C. Cao, “Some Trace Inequalities for Discrete Groups of Möbius Transformations,” Proceedings of the American Mathematical Society, Vol. 123, No3807-3815. doi:10.2307/2161910  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  F. W. Gehring and G. J. Martin, “Geodesics in Hyper-bolic 3-Folds,” Michigan Mathematical Journal, VoNo. 2, 1997, pp. 331-343. doi:10.1307/mmj/1029005708l. 44,  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