Algebraic Characterization of the Dynamical Types of Isometries of the Hyperbolic 5-space

Let M(n) denote the group of orientation-preserving isometries of H. Classically, one uses the disk model for H to define the dynamical type of an isometry. In this model an isometry is elliptic if it has a fixed point on the disk. An isometry is parabolic, resp. hyperbolic, if it is not elliptic and has one, resp. two fixed points on the conformal boundary of the hyperbolic space. If in addition to the fixed-points one also consider the “rotation-angles” of an isometry, the above classification of the dynamical types can be made finer. Let T be an isometry of H. In the linear (hyperboloid) model, every element T in M(n) has a unique Jordan decomposition T = TsTu, where Ts is semisimple, Tu unipotent. Moreover Ts and Tu are elements in M (n) and they commute. The “rotation angles” correspond to the complex conjugate eigenvalues of Ts. For each pair of complex conjugate eigenvalues {eiθ, e−iθ}, 0 < θ ≤ π, we assign a rotation-angle to T . We call T a k-rotatory elliptic (resp. k-rotatory parabolic, resp k-rotatory hyperbolic) if it is elliptic (resp. parabolic, resp. hyperbolic) with k rotation angles. For a more elaborate description we refer to [9]. From dimension three onwards this classification is finer than any of the existing classifications in the literature. A 0-rotatory parabolic (resp. a 0rotatory hyperbolic) is called a translation (resp. a stretch). Recall that in dimension 3, the group SL(2,C), or equivalently, GL(2,C) acts as linear fractional transformations of the boundary sphere S and the dynamical types are characterized by the trace [3], [16] and trace 2 det [9] respectively. In [5] Parker et al have given an