Asymptotic enumeration of reversible maps regardless of genus
Drmota, Michael ; Nedela, Roman
We derive asymptotic expansions for the numbers ▫$U(n)$▫ of isomorphism classes of sensed maps on orientable surfaces with given number of edges ▫$n$▫, where we do not specify the genus and for the ...numbers ▫$A(n)$▫ of reflexible maps with ▫$n$▫ edges. As expected the ratio ▫$A(n)/U(n) \to 0$▫ for ▫$n \to \infty$▫. This shows that almost all maps are chiral. Moreover, we show ▫$\log A(n) \sim \frac{1}{2} \log U(n) \sim \frac{n}{2} \log n$▫. Due to a correspondence between sensed maps with given number of edges and torsion-free subgroups of the group ▫$\Gamma = \langle x, y | y^2 = 1 \rangle$▫ of given index, the obtained results give an information on asymptotic expansions for the number of conjugacy classes of such subgroups of given index.
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