16:00-16:50 J Parker (Durham) Title: Classification and fixed points of quaternionic Moebius transformations Abstract: The unit ball in the quaternions is a convenient model for hyperbolic 4 dimensional space. The (orientation preserving) isometries are quaternionic Moebius transformations $g(z)=(az+b)(cz+d)^{-1}$ where $a$, $b$, $c$, $d$ are quaternions. The condition that $g$ should preserve the unit ball gives equations relating $a$, $b$, $c$ and $d$. Quaternionic Moebius transformations may be classified according to their fixed points and dynamics. We show how to use $a$, $b$, $c$ and $d$ to decide which class such a transformation belongs to and we give explicit expressions for the fixed points. Quaternionic Moebius transformations have already been used by Kellerhals to give geometric information about 4 dimensional hyperbolic manifolds. It is hoped that the information we derive will lead to further applications.