Biembeddings of Metacyclic Groups and Triangulations of Orientable Surfaces by Complete Graphs

For each integer $n\ge 3$, $n\ne 4$, for each odd integer $m\ge 3$, and for any $\lambda\in\Bbb Z_n$ of (multiplicative) order $m'$ where $m'\mid m$, we construct a biembedding of Latin squares in which one of the squares is the Cayley table of the metacyclic group $\mathbb{Z}_m\ltimes_{\lambda}\mathbb{Z}_n$. This extends the spectrum of Latin squares known to be biembeddable. The best existing lower bounds for the number of triangular embeddings of a complete graph $K_z$ in an orientable surface are of the form $z^{z^2(a-o(1))}$ for suitable positive constants $a$ and for restricted infinite classes of $z$. Using embeddings of $\mathbb{Z}_3\ltimes_{\lambda}\mathbb{Z}_n$, we extend this lower bound to a substantially larger class of values of $z$.