Hypermap operations of finite order

Abstract

Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups $\mathrm{Out}\Delta \cong PG{L}_2(\mathbf{Z})$ and $\mathrm{Out}{\Delta }^{+}\cong G{L}_2(\mathbf{Z})$ of the groups $\Delta =C_2\ast C_2\ast C_2$ and ${\Delta }^{+}=F_2$. We will consider the elements of finite order in these two groups, and the operations they induce.

arXiv comment14 pages. To appear in Discrete Mathematics (Proc. Bled Combinatorics Conf. 2007)