Advances in Pure Mathematics, 2011, 1, 218220 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* ChangJun Li, LiJie Sun, Na Li School of Mathematical Sciences, Ocean University of China, Qingdao, China Email: 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 3space is the set 33 123 3 =,, :>0HxxxRx endowed with the complete Riemannian metric d= 3 d x of constant curvature equal to –1. A Kleinian group G is a discrete nonelementary subgroup of ) 3 ( som H , where 3 () 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 tationpreserving isometry of hyperbolic 3space 3 . 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 dA . Next, for each and in we let [,] g the multiplicatie commutator 11 denotev gf g . Wree complex numbers e call the th 11 ,= 2f gtrfgfg 22 =4,=ftrf gtrg 4 the parameters of , g ice of . These parameters are inde pendent of the cho matrix representation for s and in (2, )SL Cd they determine , an, g uni quely up to conjugacy whenever ,0fg . Thelements of e of , other than thentity, fall into three types. e id 1) Elliptic: [4,0f and ) is conjugate to zz where = 1 ic: . 2) Loxodrom 4,0 d f an is conju zz gate to where=1 ; is hyperbolic if, in addition, >0 . 3) Pa rabolic: =0f nd a is conjugate to zz . If M is nonp arabolic, then fixes two points C and th points i of e closed hyperbolic line joining these two fixed s called the axis of , noted by de ax f. In this case, translates along ax f by an amount 0f , the translation lengt of h , about rotates ax f by an angle ]f , and (, 2 =4sin2 if f In [4], F.W.Gehring and G. J. Martin havehown : Theorem 1.1: [4] If s , g is discrete, if and f are loxodromics with = g , and if f ax and ax g intersect at an angle where 0< < , n the sinhsinf *The Projectsponsored 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 and hen be Möbius transforma tions, neither the identity. T and are conju gate if and only if 2 =tr ft r g 2 . Lemma 2.2: [4] If , g is a Kleinian group, if is elliptic of order 3n, and if is no of order 2, then t , gan where 2cos271if= 3 2cos25if= 4,5 =2cos26if= 6 2cos 21if7 n n an n nn Lemma 2.3: [3] Suppose that and in have disjoint pairs of fixed points in C and is hyperbolic line in the 3 which is oron of thogal to the axes and . Then 2 4, =sinh fg i fg where =,= , 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 The coefficient of sinh cannot be replaced by smaller constant. 3. Main Results heorem 3.1: If T, g is discrete, if and 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 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 , n are primitive elliptics. From Lemma 2.2 we can obtai ,3fga , so 2 sin 4 , 43 222 16sinsin sinmn 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 gfg a by 2 4, sinh fg fg To show that the inequa lity is sharp, we let , g denote the (2,3,7) triangle group where and are primitive with Copyright © 2011 SciRes. APM
C.J. LI ET AL. Copyright © 2011 SciRes. APM 220 37 f==g 2= 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 or rotation angle when is loxodromic and is loxodromic or elliptic. Theorem3.1 show the condition when and are elliptics. Corollary 3.1: If , g is discrete, if and e elliptics with ar = g , ,0fg and if ax fand ax g intersect at an angle , where 0< 2. If the order of 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 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 , g is discrete, if and are loxodromics with fg = g ax f and if f and ax g coplanar but disjoint, let ax 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 , mm 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 QiZhi Fang for her support. 5. References [1] A. F. Beardo SpringVerlag, [2] C. Cao, “Some Trace Inequalities for Discrete Groups of Möbius Transformations,” Proceedings of the American Mathematical Society, Vol. 123, No 38073815. 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. 175219. doi:10.1007/BF03008423 [4] F. W. Gehring and G. J. Martin, “Geodesics in Hyper bolic 3Folds,” Michigan Mathematical Journal, Vo No. 2, 1997, pp. 331343. 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. 3176. doi:10.1515/crll.1991.418.31
