Notes on discrete subgroups of Möbius transformations

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 123, No. 2, May 2013, pp. 245–251. c Indian Academy of Sciences 

Notes on discrete subgroups of Möbius transformations HUA WANG1 , YUEPING JIANG2 and WENSHENG CAO3 1 Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan 410076, People’s Republic of China 2 Department of Applied Mathematics, Hunan University, Changsha 410082, People’s Republic of China 3 School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, People’s Republic of China E-mail: [email protected]; [email protected]

MS received 12 January 2012; revised 27 April 2012 Abstract. Jørgensen’s inequality gives a necessary condition for a nonelementary two generator subgroup of S L(2, C) to be discrete. By embedding S L(2, C) into Uˆ (1, 1; H), we obtain a new type of Jørgensen’s inequality, which is in terms of the coefficients of involved isometries. We provide an example to show that this result gives an improvement over the classical Jørgensen’s inequality. Keywords.

Jørgensen’s inequality; quaternionic hyperbolic space; embedding.

1. Introduction Jørgensen [7] obtained the following Jørgensen’s inequality, which is important in the study of the geometry of discrete groups and the structure of 3-dimensional hyperbolic manifold. Theorem J. Let g, h ∈ S L(2, C). If the two-generator subgroup  f, g is discrete and nonelementary, then |tr2 (g) − 4| + |tr[g, h] − 2| ≥ 1, where [g, h] = ghg −1 h −1 and tr(·) is the trace function. By dealing with the geometry of the action of the group on the boundary of complex hyperbolic space, Basmajian and Miner [1] generalized the Jørgensen’s inequality to the two generator subgroup of PU (2, 1) when the generators are loxodromic or boundary elliptic. Jiang et al. [6] obtained a similar form of the above theorem in SU (2, 1). Kim [8] and Markham [9] found analogues in quaternionic hyperbolic space of results in [6]. Recently, Cao and Parker [3] and Cao and Tan [4] generalized and improved the results in [6] to the setting of quaternionic hyperbolic n-space. 245

246

Hua Wang et al.

Contrast to the above generalizations to higher dimension and bigger algebraic body, there is less attempt to improve the classical Jørgensen’s inequality. Recently, using the embedding of S L(2, C) into U (1, 1, H), Cao and Tan [4] obtained some new types of Jørgensen’s inequality with one element being elliptic. In this paper, using the embedding of S L(2, C) into Uˆ (1, 1, H), we will make another attempt to obtain a new type of Jørgensen’s inequality with one element being loxodromic. In order to compare our results with the classical Jørgensen’s inequality, we reformulate part of Theorem J as the following proposition. PROPOSITION 1.1 Let g be a loxodromic element and h be any element in S L(2, C) with the forms:     iθ 0 a b re , h= . g= c d 0 r −1 e−iθ

(1)

If g, h is discrete and non-elementary, then (r 2 + r −2 − 2 cos(2θ ))(1 + |bc|) ≥ 1.

(2)

Our main result is the following theorem. Theorem 1.1. Let g be a loxodromic element and h be any element in S L(2, C) with the forms (1). If g, h is discrete and non-elementary, then 1

|abcd| 2 ≥

1 − Mg , Mg2

where Mg = |r eiθ − 1| + |r −1 e−iθ − 1| =

(3)

 (r + r −1 + 2)(r + r −1 − 2 cos θ ). (4)

The structure of the remainder of this paper is as follows: In § 2, we give the necessary background material for quaternionic hyperbolic space and a lemma for the proof of our main result. Section 3 contains the proof of Theorem 1.1 and in § 4, we offer an example to reveal the merit of our inequality.

2. Preliminaries Let H denote the division ring of real quaternions. Elements of H have the form q = q1 + q2 i + q3 j + q4 k ∈ H, where qi ∈ R and i2 = j2 = k2 = ijk = −1. Let q¯ = q1 − q2 i − q3 j − q4 k be the conjugate of q, and   |q| = qq ¯ = q12 + q22 + q32 + q42 be the modulus of q. We define (q) = (q + q)/2 ¯ to be the real part of q, and (q) = ¯ −2 is the inverse of q. We remark (q − q)/2 ¯ to be the imaginary part of q. Also q −1 = q|q| that for a complex number c, we have jc = cj. ¯

Discrete subgroups of Möbius transformations

247

Let H1,1 be the vector space of dimension 2 over H with the unitary structure defined by the Hermitian form z, w = w∗ J z = w1 z 1 − w2 z 2 , where z and w are the column vectors in H1,1 with entries (z 1 , z 2 ) and (w1 , w2 ) respectively, ∗ denotes the conjugate transpose and J is the Hermitian matrix   1 0 J= . 0 −1 We define a unitary transformation g to be an automorphism H1,1 , that is, a linear bijection such that g(z), g(w) = z, w for all z and w in H1,1 . We denote the group of all unitary transformations by U (1, 1; H), this is sometimes also denoted by Sp(1, 1). Following § 2 of [5], let V0 = {z ∈ H1,1 − {0} : z, z = 0},

V− = {z ∈ H1,1 : z, z < 0}.

It is obvious that V0 and V− are invariant under U (1, 1; H). We define V s to be V s = V− ∪ V0 . Let P : V s → P(V s ) ⊂ H be the projection map defined by   z P 1 = z 1 z 2 −1 . z2 We define B = P(V− ), the ball model of 1-dimensional quaternionic hyperbolic space. It is easy to see that B can be identified with the quaternionic unit ball z ∈ H : |z| < 1 . Also the unit  sphere  in H is ∂B = P(V0 ). a b If g = ∈ U(1, 1; H) then, by definition, g preserves the Hermitian form. Hence c d w∗ J z = z, w = gz, gw = w∗ g ∗ J gz for all z and w in V . Letting z and w vary over a basis for V we see that J = g ∗ J g. From this we find that g −1 = J −1 g ∗ J , i.e.  −1   a¯ −c¯ a b = . c d −b¯ d¯ It is convenient to introduce the Cayley transform √ ⎞ ⎛ √ 2 2 ⎜ 2 − 2 ⎟ C =⎜ √ ⎟ ⎝ √ ⎠. 2 2 2 2

(5)

We denote Uˆ (1, 1; H) = CU (1, 1; H)C −1 , which is the automorphism group of the Hermitian form z, w = w1 z 2 + w2 z 1 .

248

Hua Wang et al.

Viewed as a projective transformation, the Cayley transformation C maps B and its bound1 and its boundary ∂ 1 , respectively. The Bergman metric ary ∂B to the Siegel domain H H 1 on H is given by the distance formula cosh2  If g =

a b c d

z, ww, z ρ(z, w) = , 2 z, zw, w 1 where z, w ∈ H , z ∈ P −1 (z), w ∈ P −1 (w).



∈ Uˆ (1, 1; H), then g −1 =



d¯ b¯ c¯ a¯

(6)

 and consequently,

¯ + bd ¯ = 0, ca a d¯ + bc¯ = 1, a b¯ + ba¯ = 0, cd¯ + d c¯ = 0, db ¯ + ac ¯ = 0. (7) 1 is defined as The quaternionic cross-ratio of four points z 1 , z 2 , w1 , w2 in H

[z 1 , z 2 , w1 , w2 ] = w1 , z1 w1 , z2 −1 w2 , z2 w2 , z1 −1 ,

(8)

where zi ∈ P −1 (z i ), wi ∈ P −1 (wi ), i = 1, 2. As in [3], the quaternionic cross-ratio [z 1 , z 2 , w1 , w2 ] depends on the choice z1 ∈ P −1 (z 1 ). However its absolute value   [z 1 , z 2 , w1 , w2 ] = |w1 , z1 w2 , z2 | |w1 , z2 w2 , z1 |

(9)

is independent of the preimage of z i and wi in H1,1 . The following lemma is crucial to us. Lemma 2.1 (cf. Theorem 1.1 of [3]). Let g be a loxodromic element of Uˆ (1, 1, H) with 1 . Let h be any other element of U ˆ (1, 1, H). If Mg < 1 and with fixed points u, v ∈ ∂H     [h(u), u, v, h(v)]1/2 [h(u), v, u, h(v)]1/2 < 1 − Mg , Mg2

(10)

then the group g, h is either elementary or not discrete.

3. Proof of Theorem 1.1 As in [2, 4, 5], we can regard Sp(1, 1) as the isometries of hyperbolic 4-space H4R , whose model is the unit ball in the quaternions H. S L(2, C), the isometries of hyperbolic 3-space H3R , can be embedded as a subgroup of Uˆ (1, 1; H) as follows: f ∈ S L(2, C) → T f T −1 ∈ U (1, 1; H)  C T f T −1 C −1 ∈ Uˆ (1, 1; H), (11)   1 −j where T = √1 and C be the Cayley transform given by (5). 2 −j 1 The following lemma can be verified directly.

Discrete subgroups of Möbius transformations

249

Lemma 3.1. Let g and h be as in (1) and gˆ = C T gT −1 C −1 , hˆ = C T hT −1 C −1 . Then   1 (1 + j)(r eiθ + r −1 e−iθ )(1 − j) (1 + j)(r eiθ − r −1 e−iθ )(1 + j) gˆ = 4 (1 − j)(r eiθ − r −1 e−iθ )(1 − j) (1 − j)(r eiθ + r −1 e−iθ )(1 + j) (12) and hˆ =



aˆ cˆ

bˆ dˆ

 ,

(13)

where 1 (1 + j)(a − c − b + d)(1 − j), 4 1 cˆ = (1 − j)(a + c − b − d)(1 − j), 4

aˆ =

1 bˆ = (1 + j)(a − c + b − d)(1 + j), 4 1 dˆ = (1 − j)(a + c + b + d)(1 + j). 4

Proof of Theorem 1.1. As in Lemma 3.1, by the embedding given by (11), we get the corresponding elements gˆ and hˆ in Uˆ (1, 1; H) of g and h. That is gˆ and hˆ are given by (12) and (13), respectively.    1+j −1 − j Let u = 12 and v = 12 . Then u and v are the preimages of the 1−j 1−j fixed points u, v of g, ˆ respectively. By direct computation, we have      = |bc|,  = |ad|, [h(u), [h(u), ˆ ˆ ˆ ˆ u, v, h(v)] v, u, h(v)]    = |bc| . [u, v, h(u), ˆ ˆ h(v)] (14) |ad| As in [3], the two right eigenvalues of gˆ can chosen to be r eiθ and r −1 e−iθ , which implies that  Mgˆ = |r eiθ − 1| + |r −1 e−iθ − 1| = (r + r −1 + 2)(r + r −1 − 2 cos θ ). ˆ we conclude the proof of Applying Lemma 2.1 to the two generator subgroup g, ˆ h, Theorem 1.1. 

4. Comparisons In this section, we will give an example to show that our result gives an improvement over the Jørgensen’s inequality. Without loss of generality, we assume r > 1 in the following case. Example 4.1.  g=

r 0 0 r −1



 ,

h=

xi yi yi xi

 , where x, y > 0, y 2 − x 2 = 1.

In this case, g and h are loxodromic and g, h is non-elementary.

(15)

250

Hua Wang et al.

In this case Mg = r − r −1 . We can assert that Theorem 1.1 is better than Jørgensen’s inequality if we can find r and x > 0 and some m = r − r1 < 1 satisfy the following inequalities:  1−m x 1 + x2 < , m2

2 + x2 ≥

1 . m2

(16)

Since the above inequalities are equivalent to  m 4 + 4(1 − m)2 − m 2 1 − 2m 2 2 , x2 ≥ , x < 2 2m m2 √ √ √ √ √ 5−1+ 22−2 5 5+1 and for 5−1 < m < 1, i.e. 1.3557 ≈ < r < ≈ 1.6180, 2 4 2 1 − 2m 2 < m2

 m 4 + 4(1 − m)2 − m 2 , 2m 2

(17)

we know that for r, x in (15) satisfying  √ √ √ 5 − 1 + 22 − 2 5 5+1