Discrete subgroups of Möbius transformations

J. Math. Anal. Appl. 397 (2013) 233–241

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Discrete subgroups of Möbius transformations✩ Huaying Huang Anhui University, School of Mathematical Sciences, No. 111, Jiulong Road, 230601 Hefei, China

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Article history: Received 5 January 2012 Available online 25 July 2012 Submitted by Richard M. Timoney

abstract In this paper, we present the necessary geometric conditions for a nonelementary two ˆ n = Rn ∪ {∞} to be discrete. generator subgroup of Möbius transformations acting in R © 2012 Elsevier Inc. All rights reserved.

Keywords: Discrete groups Möbius transformations Two generator subgroups Nonelementary subgroups

1. Introduction In the study of discrete groups it is important to find conditions for a group to be discrete. Jørgensen’s inequality gives ˆ 2 to be discrete. a necessary condition for a nonelementary two generator subgroup of Möbius transformations acting in R n ˆ Martin [1] generalized it to R . Basmajian and Miner [2] gave the necessary conditions for two generator subgroups of PU(2,1)(the projectivization of unitary group of Hermitian form of signature (2,1) on C3 ), which is the group of holomorphic isometries of complex hyperbolic space, by using the stable basin. Martin’s generalized Jørgensen’s inequality (see [1]) is not formulated in a conjugacy invariant fashion. Its geometric significance is not so obvious as the conditions given by Basmajian and Miner. Basmajian and Miner introduced the stable basin theorem. Parker improved the stable basin theorem and gave the complex hyperbolic version of Jørgensen’s inequality ([3]). In our paper, we extend the stable basin theorem of Basmajian and Miner ([2], Theorem 6.4) to Möbius transformations in all dimensions and give the necessary conditions for ˆ n to be discrete. We work in real hyperbolic a nonelementary two generator subgroup of Möbius transformations acting in R space with the Euclidean metric and get better results by comparing the data. For two generator subgroups of PU(2,1), Basmajian and Miner gave the following theorem: Theorem A (Theorem 9.1 of [2]). Suppose f , g ∈ PU (2, 1) are either loxodromic or boundary elliptic, with fixed points af , rf , ag , and rg respectively. If there exists a point (r , r , ε ) in the stable region so that

|[af , rf , ag , rg ]| < r 4 , and max{|λ(f ) − 1|, |λ(g ) − 1|} < ε, then either f and g commute, or the group generated by f and g is not discrete.

✩ Supported by Mathematics of Tianyuan Youth Fund Project (No. 11126175), the National Science Foundation of China (Grant No. 11071001).

E-mail address: [email protected]. 0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.07.029

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H. Huang / J. Math. Anal. Appl. 397 (2013) 233–241

Fig. 1. Figure of the relation of r and ε(r ).

The cross ratio mentioned in the theorem above is defined as follows:

|[q1 , q2 , q3 , q4 ]| =

d2H (q3 , q1 )d2H (q4 , q2 ) d2H (q4 , q1 )d2H (q3 , q2 )

,

where dH (·, ·) is Cygan distance. (See [2] Section 3.) ˆ n ) be the Möbius group consisting of all In this paper, we consider this problem in real space of high dimensions. Let M (R ˆ n . In PU(2,1), the complex dilation λ of an element can be written orientation-preserving Möbius transformations acting in R ˆ n ), the role of rotations are replaced by orthogonal matrices. as λ′ eiθ where λ′ > 0 and eiθ is a rotation. However in M (R Under such circumstances, the complex dilation can be considered as the composition of a real dilation and an orthogonal matrix. With this observation, we have the following result, which is the analogue of Theorem 6.4 of [2].

ˆ n ), are either loxodromic or boundary elliptic, with fixed points af , rf , ag , rg respectively, and Theorem 1.1. Suppose f , g ∈ M (R 2 [af , rf , ag , rg ] < r , 0 < r < 1. If f ∼ λA, g ∼ λ′ B, where λ, λ′ > 0, A, B ∈ O(n), and max{∥λA − I ∥, ∥λ′ B − I ∥} < ε(r ), where

ε(r ) =

1 − r2 1 + r2

.

(1.1)

Then ⟨f , g ⟩ is elementary or not discrete. In the above theorem, the cross ratio is

[af , rf , ag , rg ] =

|af − ag ||rf − rg | (see [4] Section 3.2). |af − rf ||ag − rg |

Eq. (1.1) means that (r , r , ε(r )) is a point in the stable ball region (See Section 3 and ε(r ) = ε(r , r ), where ε(r , r ′ ) is defined in Theorem 3.2.) The advantage in real space is that we use the Euclidean distance instead of Cygan distance by which the calculation is simplified and yields better results. The following Fig. 1 roughly show the relation of r and ε in the three cases. The lowest curve is the stable basin theorem given in [2], Figure 6.16. The second curve is the stable basin theorem given in [3]. The top curve is Theorem 1.1. 2. Möbius transformations

ˆ n ) can extend to an isometry of (n + 1)-dimensional hyperbolic space By Poincaré extension, we know any g ∈ M (R ˆ n++1 = {(x1 , x2 , . . . , xn , xn+1 )|(x1 , x2 , . . . , xn ) ∈ Rˆ n , xn+1 > 0} which we denote g˜ . We recall the classification of the R Möbius transformations. (See [5] Chapter 1, Section 5.)

H. Huang / J. Math. Anal. Appl. 397 (2013) 233–241

235

ˆ n ), g ̸= I is called: Definition 2.1. The transformation g ∈ M (R n +1 ˆ n+1 ) has a unique fixed point in Rˆ n++1 that lies in Rˆ n = ∂ R+ (1) parabolic if the extended transformation g˜ ∈ M (R ; n+1 ˆ n+1 ) has precisely two fixed points in Rˆ n++1 that lie in Rˆ n = ∂ R+ (2) loxodromic if g˜ ∈ M (R ; and

ˆ n+1 ) has a fixed point in Rˆ n++1 . (3) elliptic if g˜ ∈ M (R n +1 ˆ+ The extension of an elliptic element fixes either exactly one point or a totally geodesic subspace in R . We call these types regular elliptics or boundary elliptics respectively. By normalization, the following lemma is well known:

Lemma 2.2. (1) The parabolic element g is conjugate to h ◦ g ◦ h−1 (x) = Ax + b, where A ∈ O(n), b ∈ Rn \ {0} is not orthogonal to the Eigen subspace of the orthogonal transformation A; (2) The loxodromic element g is conjugate to h ◦ g ◦ h−1 (x) = λAx, where λ > 0, λ ̸= 1, and A is an orthogonal transformation; (3) The elliptic is conjugate to h ◦ g ◦ h−1 (x) = Ax, where A is an orthogonal transformation. For any n × n matrix A = (aij ), we define the Hilbert-Schmidt norm as ∥A∥ =

1/2

. Let x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ) ∈ Rˆ n . The Euclidean inner product of x and y is defined to be the real number x · y = x1 y1 + x2 y2 + · · · + xn yn . ˆ n , with respect to the Euclidean inner product, is called the Euclidean It is a positive definite inner product. The norm of x in R 1

n

i,j=0

a2ij

1

norm, denoted by |x| = (x · x) 2 = (x21 + x22 + · · · + x2n ) 2 .

ˆ n ) are loxodromic or boundary elliptic with fixed points sets {x1 , x2 }, {x3 , x4 }, Proposition 2.3. Suppose f and g in M (R respectively (If one is boundary elliptic, we choose two of its fixed points.) If necessary interchange the roles of x3 and x4 so ˆ n ) such that that the cross ratio satisfies 0 < [x1 , x2 , x3 , x4 ] < 1. Then there exist h ∈ M (R (1) h ◦ f ◦ h−1 has fixed points at 0 and ∞, and (2) h ◦ g ◦ h−1 has fixed points at distance r and

1 r

1

from 0, where r = [x1 , x2 , x3 , x4 ] 2 .

Proof. Assume h1 (x) = σ ◦ T˜ ◦ σ ◦ T (x), where T (x) = x − x1 , T˜ (x) = x − σ (x2 − x1 ). σ is the reflection in the Euclidean 1 sphere S (0, 1), then σ (x) = |xx|2 . Hence h1 ◦ f ◦ h− 1 has fixed points at 0 and ∞. 1

1

We set h2 (x) = |h1 (x3 )|− 2 · |h1 (x4 )|− 2 · x and h(x) = h2 ◦ h1 (x), then |h(x3 )| = |h2 ◦ h1 (x3 )| =



|h1 (x3 )| |h1 (x4 )|

 12

. And ˜ ˜ h1 (x3 ) = σ ◦ T ◦ σ ◦ T (x3 ) = σ ◦ T ◦ σ (x3 − x1 ) = σ (σ (x3 − x1 ) − σ (x2 − x1 )), h1 (x4 ) = σ (σ (x4 − x1 ) − σ (x2 − x1 )). And    x y   |σ (x) − σ (y)| =  2 − 2  |x| |y |     12 x y x y = − · − |x|2 |y2 | |x|2 |y2 |  12  2(x · y) 1 1 − + = |x|2 |x|2 |y|2 |y|2   12 (x − y) · (x − y) = |x|2 |y|2 =

|x − y| . |x||y|

Then by using (2.1), we have

|h(x3 )| =



|h1 (x3 )| |h1 (x4 )|

 21

 1  σ (σ (x3 − x1 ) − σ (x2 − x1 ))  2   = σ (σ (x4 − x1 ) − σ (x2 − x1 ))   1  σ (x4 − x1 ) − σ (x2 − x1 )  2  =  σ (x3 − x1 ) − σ (x2 − x1 )   |x4 −x2 |  12 =

|x4 −x1 |·|x2 −x1 | |x3 −x2 | |x3 −x1 |·|x2 −x1 |

(2.1)

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H. Huang / J. Math. Anal. Appl. 397 (2013) 233–241

 = = r.

|x1 − x3 | · |x2 − x4 | |x1 − x4 | · |x2 − x3 |

Similarly, we can get |h(x4 )| =





|h1 (x4 )| |h1 (x3 )|

1 2

 12

= 1r . So h is what we need.



In Section 4, we need the following distortion question about cross ratio concerning the action of a parabolic on pairs of points, which follows Lemma 7.3 of [2].

ˆ n ) is a parabolic element fixing ∞, q ∈ Rn , with |q| > |g (0)|. Then Lemma 2.4. Suppose g in M(R [0, q, g (0), g (q)] ≤

  |g (0)| |q| 1+ . |q| − |g (0)| |q| − |g (0)|

Proof. Since g is a parabolic fixing ∞, we assume g (x) = Ax + b, A ∈ O(n), b ∈ Rn , then |g (0)| = |g −1 (0)| and |g (q)| = |Aq + b| = | − A−1 b − q| = |g −1 (0) − q| ≥ |q| − |g (0)| > 0. Furthermore |g (q) − q| ≤ |q| + |g (q)|, then we have

|g (0)| · |g (q) − q| |g (q)| · |g (0) − q| |g (0)|(|q| + |g (q)|) ≤ (|q| − |g (0)|)|g (q)|   |g (0)| |q| ≤ 1+ .  |q| − |g (0)| |q| − |g (0)|

[0, q, g (0), g (q)] =

ˆ n ) is said to be elementary if and only if there exists a finite G-orbit in Rn+1 . Definition 2.5. A subgroup G of M (R ˆ n ) is elementary. By this definition, every Abelian subgroup of M (R 3. The stable balls In this section we introduce the definition of the stable ball and prove the stable ball region theorem which are similar to the stable basin and the stable basin theorem (Definition 3.1 and Theorem 6.4 in [2]).

ˆ n ) containing the points p and q, respectively, and they have Definition 3.1. Suppose U(p) and U(q) are open sets in (R ˆ n ). The pair (U(p), U(q)) is said to be stable with respect to F if disjoint clousures. Let F denote a set of elements in M (R for any g ∈ F , g (p) ∈ U(p), g (q) ∈ U(q). ˆ n ) : g ∼ λA, λ > 0, A ∈ O(n), |ag | < r , |rg | > Remark. We set a family F = {g ∈ M (R and repelling fixed points of g. It is clear that F is closed under conjugation by σ (x) = |xx|2 .

1 r

}, where ag , rg are attracting

c

1 Since |σ g σ (0)| = |σ g (∞)| = | g (∞) | < r implies |g (∞)| > 1r , by Definition 3.1, (Br , B 1 ) is stable with respect to F if r and only if g (0) ∈ Br for all g ∈ F , where Br = {x ∈ Rn : |x| < r }. Simulated by the stable basin theorem of Basmajian and Miner, we get the following theorem which describes how far the elements of F move the points in Rn . And this is the analogue of Theorem 6.4 of [2], so we call it the stable ball region theorem. c Theorem 3.2. Given 0 < r < 1, r ′ > 0, the pair of open sets (Br ′ , B 1 ) is stable with respect to the family F (r , ε(r , r ′ )) = r′

c {g ∈ M (Rˆ n ) : g ∼ λA, λ > 0, A ∈ O(n), ∥λA − I ∥ < ε(r , r ′ ), ag ∈ Br , rg ∈ B 1 } where ε(r , r ′ ) = r g ∈ F (r , ε(r , r )), then |g (0)| < |ag |, where ag is any fixed point of g.

Proof. Since g ∼ λA, we assume g (x) = T −1 ◦ G−1 ◦ B ◦  g ◦ B−1 ◦ G ◦ T (x), where G(x) = σ ◦  T ◦ σ (x),  T (x) = x − σ (rg − ag ), T (x) = x − ag ,  g (x) = λAx and B ∈ O(n). Then

(1−r 2 )r ′ . (1+rr ′ )r

Furthermore, if

H. Huang / J. Math. Anal. Appl. 397 (2013) 233–241

237

|g (0)| = |T −1 G−1 B gB−1 GT (0)| −1 −1 = |G B gB GT (0) − T (0)| −1  = |σ T σ B gB−1 GT (0) − σ  T −1 σ GT (0)| − 1 − 1 − 1 | T σ B gB GT (0) −  T σ GT (0)| = − 1 − 1   |T σ B gB GT (0)| · |T −1 σ GT (0)| |σ B gB−1 GT (0) − σ GT (0)| = |σ B gB−1 GT (0) − σ (ag − rg )| · |σ GT (0) − σ (ag − rg )| |ag − rg |2 · |B gB−1 GT (0) − GT (0)| = |B gB−1 GT (0) − ag + rg | · |GT (0) − ag + rg | = ≤

|ag − rg |2 · |λAB−1 GT (0) − B−1 GT (0)| |λAB−1 GT (0) − B−1 (ag − rg )| · |GT (0) − ag + rg | |ag − rg |2 · ∥λA − I ∥ · |B−1 GT (0)| . − B−1 (ag − rg )| · |GT (0) − ag + rg |

|λAB−1 GT (0)

We have GT (0) = G(−ag ) = σ T˜ σ (−ag ) = σ (σ (−ag ) − σ (rg − ag )). Therefore using (2.1) we have

|GT (0)| =

1

|σ (−ag ) − σ (rg − ag )|

=

| − ag | · |rg − ag | |ag | · |rg − ag | = . | − ag − rg + ag | |rg |

Using σ 2 = id, we have GT (0) + (rg − ag ) = σ (σ (−ag ) − σ (rg − ag )) + σ (σ (rg − ag )). Therefore using (2.1) several times we have

|GT (0) + (rg − ag )| = |σ (σ (−ag ) − σ (rg − ag )) + σ (σ (rg − ag ))| |σ (−ag ) − σ (rg − ag ) + σ (rg − ag )| = |σ (−ag ) − σ (rg − ag )| · |σ (rg − ag )| |σ (−ag )| = |σ (−ag ) − σ (rg − ag )| · |σ (rg − ag )| 1

=

=

|ag | |−ag −rg +ag | · |rg −1 ag | |ag |·|rg −ag |

|rg − ag |2 . |rg |

Using the triangle inequality and the above estimates, we have

|λAB−1 GT (0) − B−1 (ag − rg )| = |λAB−1 GT (0) − B−1 GT (0) + B−1 GT (0) − B−1 (ag − rg )| ≥ |B−1 GT (0) − B−1 (ag − rg )| − |λAB−1 GT (0) − B−1 GT (0)| ≥ |GT (0) − ag + rg | − ∥λA − I ∥ · |B−1 GT (0)| = |GT (0) + (rg − ag )| − ∥λA − I ∥ · |GT (0)| =

|rg − ag |2 |ag | · |rg − ag | − ∥λA − I ∥ · . | rg | |rg |

Then using |ag | < r and |rg | > 1/r we have

 |rg − ag | | ag | − ∥λA − I ∥ · |rg | |rg |   |ag | | ag | ≥ |rg − ag | 1 − − ∥λA − I ∥ · |rg | |rg |   2 2 ≥ |rg − ag | 1 − r − ∥λA − I ∥r .

|λAB−1 GT (0) − B−1 (ag − rg )| ≥ |rg − ag |



(3.1)

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H. Huang / J. Math. Anal. Appl. 397 (2013) 233–241

Table 1 Table of values for ε(r , r ′ ). r \ r′

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.98 0.47 0.29 0.20 0.14 0.10 0.07 0.04 0.02 0

1.94 0.92 0.57 0.39 0.27 0.19 0.13 0.08 0.04 0

2.88 1.36 0.83 0.56 0.39 0.27 0.18 0.11 0.05 0

3.81 1.78 1.08 0.72 0.50 0.34 0.23 0.14 0.06 0

4.71 2.18 1.32 0.88 0.60 0.41 0.27 0.16 0.07 0

5.60 2.57 1.54 1.02 0.69 0.47 0.31 0.18 0.08 0

6.48 2.95 1.75 1.15 0.78 0.53 0.34 0.20 0.09 0

7.33 3.31 1.96 1.27 0.86 0.58 0.37 0.22 0.10 0

8.17 3.66 2.15 1.39 0.93 0.62 0.40 0.24 0.10 0

9.00 4.00 2.33 1.50 1.00 0.67 0.43 0.25 0.11 0

The last inequality in (3.1) implies that when ∥λA − I ∥r 2 < 1 − r 2 , we have

∥λA − I ∥ · |GT (0)| · |rg − ag |2 (|GT (0) + (rg − ag )| − ∥λA − I ∥ · |GT (0)|) · |GT (0) + (rg − ag )|

|g (0)| ≤

∥λA − I ∥ · ≤

|ag |·|rg −ag | |rg |

· |rg − ag |2

|rg − ag |(1 − r 2 − ∥λA − I ∥r 2 )

| r g −a g | 2 |rg |

∥λA − I ∥ · |ag | 1 − r 2 − ∥λA − I ∥r 2 ∥λA − I ∥r . ≤ 1 − r 2 − ∥λA − I ∥r 2

≤

(3.2)

Therefore, to ensure |g (0)| < r ′ it suffices to find ∥λA − I ∥ so that

∥λA − I ∥r < r ′ and ∥λA − I ∥r 2 < 1 − r 2 . 1 − r 2 − ∥λA − I ∥r 2 This is equivalent to

∥λA − I ∥ < min



 (1 − r 2 )r ′ 1 − r 2 , . (1 + rr ′ )r r2

Observe that 1 − r2 (1 − r 2 )r ′ < . (1 + rr ′ )r r2 (1−r 2 )r ′ So we set ε(r , r ′ ) = (1+rr ′ )r , then if ∥λA − I ∥ < ε(r , r ′ ), we have |g (0)| < r ′ . That is, if g ∈ F (r , ε(r , r ′ )), then |g (0)| < r ′ . 2

r If g ∈ F (r , ε(r , r )), then ∥λA − I ∥ < ε(r , r ) = 11− , then using (3.2), we have +r 2

|g (0)| ≤

∥λA − I ∥ · |ag | 1 − r 2 − ∥λA − I ∥r 2

for all g ∈ F (r , ε(r , r )).

< |ag |,



Remark. To clarify the relationship between r , r ′ , ε(r , r ′ ), we introduce some terminologies. We call a triple of non-negative numbers (r , r ′ , ε ) is a ball point provided 0 < r < 1 and 0 < ε < ε(r , r ′ ) and the triple (r , r ′ , ε ) a stable ball point. Call the set of all such points the stable ball region. We present the following numerical approximations to ε(r , r ′ ) (see Table 1). 2

r If r ′ = r, we set ε(r ) = ε(r , r ) = 11− . +r 2 We denote the order of A ∈ O(n) by o(A), then we have the following result. c

Theorem 3.3. Given 0 < r < 1, r ′ > 0, m ≥ 1, the pair of open sets (Br ′ , B 1 ) is stable with respect to the family r′

c

ˆ n ) : g ∼ λA, λ > 0, A ∈ O(n), o(A) = m, |λm − 1| < ε(r , r ′ ), ag ∈ Br , rg ∈ B 1 }, where F (r , ϵ(r , r ′ )) = {g ∈ M (R ε(r , r ′ ) =

(1−r 2 )r ′ . (1+rr ′ )r

r

Furthermore, if g ∈ F (r , ε(r , r )), then |g m (0)| < |ag |, where ag is any fixed point of g.

H. Huang / J. Math. Anal. Appl. 397 (2013) 233–241

239

Proof. Similar to the proof of Theorem 3.2, we have g (x) = T −1 G−1 B gB−1 GT (x), where B ∈ O(n), T (x), G(x) and  g is defined in Theorem 3.2. Since o(A) = m,  g m (x) = λm x. Then g m (x) = T −1 G−1 g m GT (x) and we get

|g m (0)| =

|ag − rg |2 | g m GT (0) − GT (0)| | (0) − ag + rg ||GT (0) − ag + rg | g m GT

|ag − rg |2 |λm GT (0) − GT (0)| |λ (0) − ag + rg ||GT (0) − ag + rg | |λm − 1|r ≤ . 1 − r 2 − |λm − 1|r 2

=

m GT

(1−r 2 )r ′

If |λm − 1| < ε(r , r ′ ) = (1+rr ′ )r , then |g m (0)| < r ′ . Furthermore, replacing r ′ by |ag |, where ag is any fixed point of g, we have |g m (0)| < |ag |, if g ∈ F (r , ε(r , r )).



4. The main results We discuss the case when two elements are loxodromic or boundary elliptic first, which is the analogue of Theorem 9.1 of [2].

ˆ n ) is loxodromic and g ∈ M (Rˆ n ) is loxodromic Theorem 4.1 (Loxodromic–Loxodromic or Boundary Elliptic). Suppose f ∈ M (R or boundary elliptic, with fixed points af , rf , ag , rg , respectively. If f ∼ λA, g ∼ λ′ B, λ, λ′ > 0, A, B ∈ O(n) and there exists a point (r , r , ε) in the stable ball region such that [af , rf , ag , rg ] < r 2 and max{∥λA − I ∥, ∥λ′ B − I ∥} < ε, then ⟨f , g ⟩ is elementary or not discrete. Proof. We denote the set of fixed points of f by fix(f ). And if g is boundary elliptic, we choose two fixed points of g and label them as the attractor and the repeller such that they satisfy the cross ratio condition. Suppose ⟨f , g ⟩ is not elementary, then fix(f ) ∩ fix(g ) = ∅, so [af , rf , ag , rg ] > 0. By normalization, we can assume the two fixed points of f are 0 and ∞. And by the cross ratio condition [af , rf , ag , rg ] < r 2 and Proposition 2.3, we have ag ∈ Br c

and rg ∈ B 1 . r

Consider the sequence of f -conjugates g1 = gfg −1 , g2 = g1 fg1−1 , . . . , gn = gn−1 fgn−−11 , . . . where gi are loxodromic, i = 1, 2, . . .. We claim the sequence{gn } are distinct and contained in F (r , ε). c c For ag ∈ Br , rg ∈ B 1 , g ∼ λA and ∥λA − I ∥ < ε , and (r , r , ε) is in the stable ball region, by Theorem 3.2, (Br , B 1 ) is stable r

r

c

with respect to the family F (r , ε), which means g (0) ∈ Br , g (∞) ∈ B 1 . Hence g (∞) ̸= 0, and furthermore, |g (0)| < |ag |. r

Since fix(f ) ∩ fix(g ) = ∅, g (0) ̸= 0. c For g (0), g (∞) are the two fixed points of g1 , we denote g (0) by ag1 and g (∞) by rg1 , then ag1 ∈ Br , rg1 ∈ B 1 , g1 ∼ λA c

r

and ∥λA − I ∥ < ε , by Theorem 3.2, |g1 (0)| < |ag1 |, g1 (∞) ∈ B 1 , hence g1 (∞) ̸= 0. r

We claim g1 (0) ̸= 0. If g1 (0) = 0, that is, gfg −1 (0) = 0, then f (g −1 (0)) = g −1 (0), that is, g −1 (0) is a fixed point of f . And for f being loxodromic with fixed points 0 and ∞, we get g −1 (0) = 0 or g −1 (0) = ∞, that is, g (0) = 0 or g (∞) = 0, which is a contradiction. For g1 (0), g1 (∞) ∈ fix(g2 ), we denote g1 (0) by ag2 and g1 (∞) by rg2 . Since |ag2 | = |g1 (0)| < |ag1 | and g1 (0) ̸= 0, then ag2 ∈fix(g1 ). Hence g1 and g2 are distinct. Now assume that g1 , g2 , . . . , gn ∈ F (r , ε) are distinct with gi ∼ λA and with fixed points agi+1 = gi (0), rgi+1 = gi (∞) in fix(gi ), i = 1, 2, . . . , n, and |agn | < |agn−1 | < · · · < |ag1 |. Then by the stable ball region theorem, |agn+1 | = |gn (0)| < |agn |, c

rgn+1 ∈ B 1 and gn+1 ∼ λA is still loxodromic, hence gn+1 ∈ F (r , ε), then fix(gn+1 ) is distinct from fix(gi ), i = 1, 2, . . . , n, r

hence {gn } ⊂ F (r , ε) are distinct. c Since Br and B1/r have disjoint compact closures, there exists a sequence {gnk } such that {agnk } → a∞ , and {rgnk } → r∞ ̸= a∞

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For gnk ∼ λA and with fixed points agnk , rgnk , then gnk has the following form −1 1 gnk (x) = h− nk ◦ λCnk ACnk ◦ hnk (x),

where hnk (x) = σ  Tnk σ Tnk (x),  Tnk (x) = x − σ (rgnk − agnk ), Tnk (x) = x − agnk and Cnk ∈ O(n). For there exists a sequence {Cnk } ⊆ {Cnk } such that l

lim Cnk = C ∈ O(n),

l→∞

l

hence T∞ σ T∞ (x), Tnk σ Tnk (x) = σ  lim hnk (x) = lim σ 

l→∞

l

l

l

l→∞

−1 −1 T∞ σ T∞ ◦ λCAC −1 ◦ σ  T∞ σ T∞ (x), where  T∞ = x − σ (rg∞ − ag∞ ) and T∞ (x) = x − ag∞ . Hence gnk → g∞ , where g∞ (x) = σ  l which is g∞ ∼ λA, λ > 0, A ∈ O(n). Therefore ⟨f , g ⟩ is not discrete. 

On the other hand, we consider the case when one is parabolic and the other is loxodromic, and the following theorem is the analogue of Theorem 9.11 of [2]. Theorem 4.2 (Parabolic-loxodromic). Fix a stable ball point (r , r , ε). Let g be a parabolic with fixed point ∞. If f ∼ λA is a loxodromic with attracting fixed point 0 and the repelling fixed point q satisfying ∥λA − I ∥ < ε and

|q| ≥

|g (0)|  r2

1 + r2 +





1 + r2 ,

(4.1)

then ⟨f , g ⟩ is not discrete. We need the following lemma to prove Theorem 4.2.

ˆ n ) and g1 be loxodromic. If g1 and g2 have precisely one common fixed point, then the group Lemma 4.3. Let g1 , g2 ∈ G ⊂ M (R G is not discrete. Proof. Without loss of generality, we assume that g1 (0) = 0, g2 (0) ̸= 0 and g1 (∞) = g2 (∞) = ∞. Then g1 (x) = α U1 (x), g2 (x) = β U2 (x) + b, where α, β > 0, 0 < α < 1, b ∈ Rn \ {0}, U1 , U2 ∈ O(n). We consider the following elements in the group G: hk (x) = g2 g1k g2−1 g1−k (x) = U2 U1k U2−1 U1−k (x) − α k U2 U1k U2−1 (b) + b. If U1 is finite order with U1m = I , m ∈ N. We take k = mj as multiples of m. If not, there exists a sequence {mj }, such that mj

limj→∞ U1 = I. Because 0 < α < 1, lim hmj (x) = x + b.

j→∞

Thus G is not discrete.



Proof of Theorem 4.2. If q = ∞ then by the Lemma 4.3, ⟨f , g ⟩ is not discrete. Note (4.1) implies that |q| > |g (0)|. By Lemma 2.4,

[0, q, g (0), g (q)] ≤

  |q| |g (0)| 1+ . |q| − |g (0)| |q| − |g (0)|

(4.2)

By (4.1), the right side of (4.2) is less than or equal to r 2 . That is, (4.1) implies

[0, q, g (0), g (q)] ≤ r 2 . Now set h = gfg −1 . Note that the fixed points of h are g (0) and g (q),and h ∼ λA. Moreover for the fixed points of f and h are distinct, and hence they do not commute. Thus by Theorem 4.1, ⟨f , h⟩ is not discrete, hence ⟨f , g ⟩ is not discrete.  Finally, suppose one of our generators is regular elliptic, denoted by g. We denote the extended transformation of regular elliptic g by  g, then  g has two fixed points in Rn+1 , thus  g is a boundary elliptic in M (Rn+1 ). If f is loxodromic, parabolic or boundary elliptic, the extended transformation of f is of the same type as f . Thus if one of the generators is regular elliptic, we consider their extended transformations, then we can apply the above discussions. Acknowledgments This work is supervised by Prof. Chen Min. I am grateful to Prof. Chen for his patient instructions. I thank the referee for a most careful reading of the paper and numerous comments, corrections and valuable suggestions which greatly improved it.

H. Huang / J. Math. Anal. Appl. 397 (2013) 233–241

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