%0 Journal Article
%T Inequalities for Discrete M£¿bius Groups in Infinite Dimension
%A Xi Fu
%J Journal of Complex Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/515109
%X We establish some inequalities for discrete M£¿bius groups in infinite dimension, which are generalizations of the corresponding results of Coa, 1996 and Gehring and Martin, 1991 in finite dimension. 1. Introduction Let denotes the group of all M£¿bius transformations in and the subgroup of consisting of all orientation-preserving M£¿bius transformations. The chordal distance between two points is defined by (see [1]) 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 [2] 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 [3]); see [4] for the general cases. In 1991, Frunz£¿ [5] introduced the concept of M£¿bius transformations in infinite dimension and discussed the Clifford matrix representations of M£¿bius transformations in infinite dimension. Recently, by using the representations, Li [6¨C9] discussed the properties of M£¿bius transformations in infinite dimension and obtained several analogous J£¿rgensen¡¯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. 2. Preliminaries 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
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