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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.