Description: |
Regular maps are triples $(G,a,b)$ consisting of a group $G$ and a pair $(a,b)$ of generators of $G$ (in fact a conjugacy class) being the product $ab$ an involution. A regular map ${\cal M}=(G,a,b)$ is reflexible or chiral according as $\cal M$ is isomorphic or not to its mirror image $\overline{{\cal M}}=(G,a^{-1},b^{-1})$. There is a widespread conviction that the ratio of the number of regular reflexible maps with size $n$ (order of the group $G$) over the number of regular chiral ones with same size is asymptotically zero. This seems to be supported by a calculation on the Suzuki groups by Dimitri and Hubard. In this talk we show that in the restricted families of regular maps with $p$ prime ``faces" (i.e. orbits of $a$) one can have different, and sometimes surprising, results. For instance, following a recent work by Maria Elisa Fernandes and myself, we show that for certain primes $p$ the ratio $\RoC_p(n)$, of the number of reflexible regular oriented maps with $p$ faces up to size $pn$ over the number of chiral regular oriented maps with $p$ faces up to size $pn$ ($p>3$ and $n\geq p-1$), has limit when $n\to\infty$, and depending on the classes of primes, this limit can be $1$, greater than $1$ or less than $1$.
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