Speaker
Dr
Adnan Melekoglu
(Adnan Menderes University)
Description
It is known that every Riemann surface of genus $g$ can be expressed in the form
$\mathbb{U}/\Omega$, where $\mathbb{U}$ is the Riemann sphere $\Sigma$,
the Euclidean plane $\mathbb{C}$, or the hyperbolic plane $\mathbb{H}$,
depending on whether $g$ is $0$, $1$ or $>1$, respectively, and $\Omega$
is a discrete group of isometries of $\mathbb{U}$.
A Riemann surface $S=\mathbb{U}/\Omega$ is \emph{quasi-Platonic} if $\Omega$
is normal in the ordinary triangle group $\Gamma[l,m,n]$. So $S$ underlies
a regular hypermap $\mathcal{H}$ of type $(l,m,n)$. If $\Omega$ is also
normal in the extended triangle group $\Gamma(l,m,n)$, then $\mathcal{H}$
is reflexible. Each reflection of $\mathcal{H}$ fixes a number of simple
closed geodesics on $S$, which are called \emph{mirrors}. Then every mirror
passes through some geometric points of $\mathcal{H}$ and these geometric
points form a periodic sequence, which is called the \emph{pattern} of the
mirror. By geometric points we mean the centers of the hypervertices,
hyperedges and hyperfaces of $\mathcal{H}$.
In previous work David Singerman and I classified the patterns of mirrors
on Platonic surfaces, which underlie regular maps, and in this work it is
generalized to quasi-Platonic surfaces.
Primary author
Dr
Adnan Melekoglu
(Adnan Menderes University)
