Research Article | Open Access
Inequalities for Discrete Möbius Groups in Infinite Dimension
We establish some inequalities for discrete Möbius groups in infinite dimension, which are generalizations of the corresponding results of Coa, 1996 and Gehring and Martin, 1991 in finite dimension.
Let denotes the group of all Möbius transformations in and the subgroup of consisting of all orientation-preserving Möbius transformations. The chordal distance between two points is defined by (see ) where is the stereographic projection from to the unit sphere in . The chordal metric on is defined by By using this metric and Clifford algebra, Gehring et al. established a series of inequalities for discrete subgroups of . For example, Gehring and Martin  proved that if is a discrete nonelementary subgroup of , then ; when , by using Clifford algebra, Cao generalized it to and obtained that if is discrete nonelementary subgroup of and is hyperbolic, then (see ); see  for the general cases. In 1991, Frunză  introduced the concept of Möbius transformations in infinite dimension and discussed the Clifford matrix representations of Möbius transformations in infinite dimension. Recently, by using the representations, Li [6–9] discussed the properties of Möbius transformations in infinite dimension and obtained several analogous Jørgensen’s inequalities in . In this paper, we continue the study in this direction and some new inequalities for discrete subgroups of are established. In Section 2, we give the preliminaries. The main results are given in Section 3.
The Clifford algebra is the associative algebra over the real field , generated by a countable family subject to the following relations: and no others. Every element of can be expressed by the following type: where , , , is a fixed natural number depending on , are the coefficients, and . If , then is called the real part of and denoted by ; the remaining part is called the imaginary part of and denoted by . For each , the Euclidean norm is expressed by
The algebra has three important involutions.(1) “”: replacing each of by , we get a new element . is an isomorphism of : for .(2) “”: replacing each of by . is an anti-isomorphism of : (3) “−": . It is obvious that is also an anti-isomorphism of .
For elements of the following type: we call them vectors. The set of all such vectors is denoted by and we let . For any , we have and . For , the inner product of and is given by where , .
It is easy to verify that any nonzero vector is invertible in with . The inverse of a vector is invertible too. Since any product of nonzero vectors is invertible, we conclude that any product of nonzero vectors is invertible in . The set of products of finitely many nonzero vectors is a multiplicative group, called Clifford group and denoted by .
If a matrix satisfies the following:(1);(2);(3), , , ,
then we call a Clifford matrix in infinite dimension; the set of all such matrices is denoted by . The norm of is
Obviously, ; that is, is the inverse of . By a simple computation, we know that is a multiplicative group of matrices.
For any , the corresponding mapping is a bijection of onto itself, which we call a Möbius transformation in infinite dimension. Correspondingly, the set of all such mappings is also a group, which is still denoted by .
Now, we give a classification to elements of as follows. A nontrivial element is called: (1)loxodromic if it is conjugate in to , where , , and ; if , then is called hyperbolic;(2)parabolic if it is conjugate in to , where , , , and ; if , then is called strictly parabolic; if there is a positive integer such that is strictly parabolic and for any positive integer , is not strictly, then is called -strictly parabolic;(3)elliptic if is neither loxodromic nor parabolic.
For a subgroup , we say that is discrete if and only if and imply that for all sufficiently large . Otherwise, is nondiscrete.
The following is a necessary condition for a two-generator discrete subgroup of in which one generator is strictly parabolic.
Lemma 1 (see ). Suppose that is a discrete subgroup of SL and are represented by where . Then
Let , the chordal distance between them is defined as follows (see ): The chordal metric on is set to be We call the chordal norm of . It is easy to see that . The readers can refer to [5, 6, 8–10] for more details about Möbius groups in infinite dimension.
3. The Main Results and Proofs
Now we come to state and prove our main results.
Theorem 2. Suppose that is a discrete subgroup of SL and are represented by where . If , then
Proof . Let . Then This proves (19). For (20), we let , . Since we have So (20) follows. For (21), we consider the Shimizu-Leutbecher sequence Then It follows that Suppose that . We obtain, by induction, So, The above relation and (26) imply that Now Thus, by induction, Since is discrete and we have for sufficiently large . Hence . It is a contradiction. Therefore, .
By a simple caculation, we have the following.
Corollary 3. Suppose that is a discrete subgroup of SL. If is hyperbolic with and , then
Theorem 4. Suppose that is a discrete subgroup of SL and are represented by If is m-strictly parabolic and , then where .
Corollary 5. Suppose that is discrete in SL and are represented by where is strictly parabolic. If , then
Remark 6. Following [6–8], we know that there are much differences between subgroups of and such as the discreteness and the fixed points sets of elements, which play key roles in the proofs of [3, Theorem 3.11] and [2, Theorem 6.14]. This implies that our results cannot be deduced directly by the same methods used in [2, 3].
The research was partly supported by Tian-Yuan Foundation (no. 11226096).
- A. F. Beardon, The Geometry of Discrete Groups, vol. 91 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1983.
- F. W. Gehring and G. J. Martin, “Inequalities for Möbius transformations and discrete groups,” Journal für die Reine und Angewandte Mathematik, vol. 418, pp. 31–76, 1991.
- C. Cao, “The chordal norm of discrete Möbius groups in several dimensions,” Annales Academiae Scientiarum Fennicae Series A, vol. 21, no. 2, pp. 271–287, 1996.
- G. J. Martin, “On discrete Möbius groups in all dimensions: a generalization of Jørgensen's inequality,” Acta Mathematica, vol. 163, no. 3-4, pp. 253–289, 1989.
- M. Frunză, “Möbius transformations in infinite dimension,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 36, no. 7-8, pp. 369–376, 1991.
- L. Li, “Ball-preserving Möbius transformations in infinite dimension,” Complex Variables and Elliptic Equations, vol. 54, no. 7, pp. 697–703, 2009.
- L. Li, “A generalization of Jørgensen's inequality to infinite dimension,” New York Journal of Mathematics, vol. 17, pp. 41–49, 2011.
- L. Li, “Discreteness of Möbius groups in infinite dimension,” Complex Variable, vol. 58, no. 1, pp. 109–112, 2013.
- L. L. Li, “An inequality on parabolic Möbius groups in infinite dimension and their application,” Acta Mathematica Scientia Series A, vol. 32, no. 2, pp. 349–355, 2012.
- L. Li, “A class of Möbius transformations in infinite dimension,” Journal of Jiangxi Normal University (Natural Sciences Edition), vol. 33, pp. 556–559, 2009.
- F. W. Gehring and G. J. Martin, “The matrix and chordal norms of Möbius transformations,” in Complex Analysis, pp. 51–59, Birkhäuser, Basel, Switzerland, 1988.
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