### Maximal order group actions on Riemann surfaces

#### Abstract

A natural problem is to determine, for each value of the integer *g* ≥ 2, the largest order of a group that acts on a Riemann surface of genus *g*. Let *N*(*g*) (respectively *M*(*g*)) be the largest order of a group of automorphisms of a Riemann surface of genus *g* ≥ 2 preserving the orientation (respectively possibly reversing the orientation) of the surface.

The basic inequalities comparing *N*(*g*) and *M*(*g*) are *N*(*g*) ≤ *M*(*g*) ≤ 2*N*(*g*). There are well-known families of extended Hurwitz groups that provide an infinite number of integers *g* satisfying *M*(*g*) = 2*N*(*g*). It is also easy to see that there are solvable groups which provide an infinite number of such examples.

We prove that, perhaps surprisingly, there are an infinite number of integers *g* such that *N*(*g*) = *M*(*g*). Specifically, if *p* is a prime satisfying *p* ≡ 1 (mod 6) and *g* = 3*p* + 1 or *g* = 2*p* + 1, there is a group of order 24(*g* − 1) that acts on a surface of genus *g* preserving the orientation of the surface. For all such values of *g* larger than a fixed constant, there are no groups with order larger than 24(*g* − 1) that act on a surface of genus *g*.

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MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2257.6de

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications