Hurwitz's automorphisms theorem

Hurwitz's automorphisms theorem is a theorem about the automorphisms of compact Riemann surfaces of genus ≥ 2, with implications for regular maps.

Statement

Regular maps

An orientable regular map, of genus $g\geq 2$, can have at most $168(g-1)$ flags. A non-orientable regular map, of $g\geq 2$, can have at most $84(g-1)$ flags

Automorphism groups

The automorphism group, $\mathrm {Aut} (X)$, of a compact Riemann manifold, X , is the group of orientation-preserving conformal mappings from X  to X . For a smooth connected Riemann manifold of genus g  ≥ 2, its automorphism group, $\mathrm {Aut} (X)$, has at most order $84(g-1)$.

Examples

For regular maps of genus 0 and 1 the Hurwitz bound does not apply. Regular maps of these genera can can have arbitrarily high flag counts. For example in n -gonal duocombs are genus 1 maps with $8n^{2}$ flags.

Examples of regular maps with maximal flags

Genus	Hurwitz bound	Example regular map	Orientation-preserving automorphisms	Flag count	Fundamental domain
2	84	P(8,3)	48	96
3	168	Klein map	168	336
4	252	Dodecadodecahedron	120	240
5	336	Fricke-Klein map	192	384

External links

• Wikipedia contributors. "Hurwitz's automorphisms theorem".