Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes |
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Authors: | Wensheng Cao Krishnendu Gongopadhyay |
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Institution: | (1) Department of Mathematics and Computer Science, University of Haifa, Mount Carmel, Haifa, 31905, Israel |
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Abstract: | Let
H2\mathbb F{{\bf H}^{\bf 2}_{\mathbb F}} denote the two dimensional hyperbolic space over
\mathbb F{\mathbb F} , where
\mathbb F{\mathbb F} is either the complex numbers
\mathbb C{\mathbb C} or the quaternions
\mathbb H{\mathbb H} . It is of interest to characterize algebraically the dynamical types of isometries of
H2\mathbb F{{\bf H}^{\bf 2}_{\mathbb F}} . For
\mathbb F=\mathbb C{\mathbb F=\mathbb C} , such a characterization is known from the work of Giraud–Goldman. In this paper, we offer an algebraic characterization
of isometries of
H2\mathbb H{{\bf H}^{\bf 2}_{\mathbb H}} . Our result restricts to the case
\mathbb F=\mathbb C{\mathbb F=\mathbb C} and provides another characterization of the isometries of
H2\mathbb C{{\bf H}^{\bf 2}_{\mathbb C}} , which is different from the characterization due to Giraud–Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The z-classes provide a finite partition of the isometry group. In this paper, we describe the centralizers of isometries of
H2\mathbb F{{\bf H}^{\bf 2}_{\mathbb F}} and determine the z-classes. |
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Keywords: | |
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