The background

The present list contains the census of discrete actions of finite groups on Riemann surfaces of genus \(2\leq g \leq 21\). The page presents the accompanying data for the paper Computing equivalence classes of finite group actions on orientable surfaces [KNS].

A finite group \(G\) has an orientation-preserving action on an orientable surface \(\mathcal{S}_g\), if there exist a Fuchsian group \(\Gamma\), acting as a group of orientation-preserving homeomorphisms of the universal cover \(\mathcal{U}\) of \(\mathcal{S}_g\), and a torsion-free normal subgroup \(K\unlhd\Gamma\) such that \(G\cong\Gamma/K\). The projection \(\eta\colon \Gamma\to \Gamma/K\) corresponds to the regular covering \(\mathcal{U}\to\mathcal{S}_g\), see e.g. [Br22].

The standard presentation of group \(\Gamma = F(x_1,x_2,\ldots,x_r,a_1,b_1,\ldots,a_\gamma,b_\gamma)\) reads as follows \[ \langle x_1,x_2,\ldots,x_r,a_1,b_1,\ldots,a_\gamma,b_\gamma\ \mid\ x_1^{m_1}=x_2^{m_2}=\cdots=x_r^{m_r}=1, \prod_{j=1}^\gamma [a_j,b_j]\prod_{i=1}^r x_i=1\rangle. \] The group \(\Gamma\) is fully determined by its signature written in the form \((\gamma;m_1,m_2,\ldots,m_r)\), where \(\gamma\) is the genus of orientable quotient orbifold \(\mathcal{O}=\mathcal{S}_g/G\) and for all \(i=1,\ldots,r\), the integers \(m_i>1\) determine the indices of the corresponding branch points. These points are images of \(r\) singular points in the branched covering \(\kappa\colon\mathcal{S}_g\to\mathcal{O}\), induced by the action of \(G\) on \(\mathcal{S}_g\). Given \(\kappa\), the parameters of the presentation of \(\Gamma\) are related by the Riemann-Hurwitz formula \[ 2-2g = |G|\left[2-2\gamma-\sum_{i=1}^r\left(1-\frac{1}{m_i}\right)\right], \] where \(\gamma \leq g\) is the genus of the orbifold \(\mathcal{O}\), \(|G|\) is the order of the group \(G\), and \(m_i, i=1,2,\ldots,r\), are non-trivial divisors of \(|G|\). In particular, if \(g>1\), we have \(|G|\leq 84(g-1)\) and therefore, there are finitely many group actions on a surface \(\mathcal{S}_g\). By Riemann existence theorem, an action of \(G\) on $S_g$ with signature \(\sigma=(\gamma;m_1,...,m_r)\) exists if and only if there exists an order-preserving (smooth) epimorphism \(\eta\colon \Gamma\to G\), where the signature \(\sigma\) determines the Fuchsian group \(\Gamma\). The epimorphism \(\eta\) is order-preserving if for every (elliptic) genenerator \(x_i\) of \(\Gamma\) holds \(|\eta(x_i)|=m_i\), \(i=1,\ldots,r\). Ther pair \((\Gamma, G)\) is called \(g\)-admissible if such epimorphism exists. It is well known that a \(g\)-admissible pair \((\Gamma, G)\) may correspond to many actions of \(G\) on \(\mathcal{S}_g\).

On the other hand, the action of the group \(G\) on the surface \(\mathcal{S}_g\) is fully determined by the vector of length \((r+2\gamma)\) of images of generators of \(\Gamma\) in \(\eta\) in the form \[(\eta(x_1),\eta(x_2),\ldots,\eta(x_r),\eta(a_1),\eta(b_1),\ldots,\eta(a_\gamma),\eta(b_\gamma)).\]

By [Lloyd], two actions \(\eta_1\) and \(\eta_2\) are topologically equivalent[Br22] (shortly equivalent), if and only there exist an (orientation-preserving) automorphism \(\alpha\in\operatorname{Aut}^+\Gamma\) and an automorphism \(a\in\operatorname{Aut} G\), such that \[ \eta_2 = a\eta_1\alpha, \] that means \[ \eta_2(y) = a(\eta_1(\alpha(y))),\\ \] for all \(y\in\Gamma\). If one decides to consider the automorphism \(\alpha\) to be the identity, the corresponding equivalence have still meaning and it is related with automorphisms of surface coverings and with symmetries of discrete (algebraic) objects such as maps, hypermaps, or tesselations, see e.g. [Co1, Co2, Co3, KN12]. This equivalence is called \(\operatorname{Aut}(G)\)-equivalence (denoted also autg) and we computed the censa with respect to this equivalence in earlier versions (some of the lists are included here).

The algorithm computes exact equivalence classes of actions in case where \(\Gamma\) is planar [ZVC] or it is a surface group. This part of the algorithm is based on the papers of Tap [Tap88] and McCool [Mc96], respectively. In the general case we compute a refinement of the topological equivalence. This part is based on the results of Harvey [Ha71] and others. It follows that in general we obtain only an upper bound of the number of equivalence classes. However, we did not find an example of a g-admissible pair \((\Gamma, G)\), for which this approximation differs from the topological equivalence! For all the details, please read the accompanying paper [KNS].

Other censa of actions of finite group actions on orientable surfaces were published: genera \(2\) and \(3\) were treated by Broughton [Br91], genus \(4\) has been done by Bogopolsky [Bo97]. The existence of particular actions of genus \(5\) has been established in [KK90]. The existence of actions up to genus \( 48\) has been done by Breuer [Bre]. The online catalogue of actions of finite groups has been also published online by Paulhus [PaW]. Unfortunately, these results contain inconsistencies (except [Bo97]). For detailed discussion see [Section 8, KNS].

The census

We were able (August 15th, 2023) to determine completely the equivalence classes with respect to topological equivalence up to genus \(9\) [KNS].

For genus \(2 \leq g \leq 8\), our algorithm produces a complete set of representatives of classes of topological equivalence of finite groups actions on \(\mathcal{S}_g\) with planar signatures and with signatures of the form \((\gamma;-)\). If both \(r > 0\) and \(\gamma > 0\), the algorithm determines orbits of the action of \(\operatorname{Aut}^+(G) \times H\) on \(\operatorname{Epi}_o(\Gamma, G)\) where \(H \leq \operatorname{Aut}^+(\Gamma)\). The corresponding actions were treated case by case in [Section 7, KNS]. For genus \(g=9\) the \(g\)-admissible pairs needed to be checked are: O9.10, O9.15, O9.18, O9.20, O9.26, O9.50, O9.51, O9.52, O9.54, O9.69, O9.102, O9.104, and O9.109. By using similar methods as in [Section 7, KNS] we have confirmed the correctness of the output of the algorithm, except in case O9.109. In this case the algorithm gives two representatives of equivalence classes, however, we were not able to prove whether they are topologically equivalent or not (actualised Thu Jan 18 10:19:18 CET 2024).

As concerns genera \(g>9\) we computed \(g\)-admissible pairs and \(\operatorname{Aut}(G)\)-classes (where possible). If the number of representatives of \(\operatorname{Aut}(G)\)-equivalence is known (see the parameter autg e.g. in the corresponding index), then the list contains the exact list of representatives. Otherwise, the list of vectors is (just) reduced with respect to \(\operatorname{Aut}(G)\)-equivalence.

We provide an export program in Python (json-export.py) which can be used to produce an index file (as presented here), or a full (text human-readable) file, where the actions are described as permutations, or a \(\mathrm{\LaTeX}\) table. The export to Magma[Magma] and/or to GAP[GAP21] is planned. You can download the export program here. Please note that for running the program you'll need Python 3 (3.x). Leave the file small.dic in the same directory as json-export.py. Consult the help (python3 json-export.py -h) for furher details.

JSON data file has the following format. It contains some description in keys version, created, exported, which are not so important in principle. The key actions denotes (contains) a list of representatives with respect to given equivalence. Here is the description of the record. It contains keys:

• genus, the genus of the surface,
• gamma, the genus of the quotient orbifold,
• bd, the list (a multiset) of branch-indices,
• group a small group library identification code [BEB], consisting of
  • the order order (\(=n\))
  • the index code (\(= k\))
• autgreps is the list of representatives of \(\operatorname{Aut}(G)\)-equivalence classes. The list consists of vectors of permutations of length \(r+2\gamma\); each vector is of the form \[(\eta(x_1),\eta(x_2),\ldots,\eta(x_r),\eta(a_1),\eta(b_1),\ldots,\eta(a_\gamma),\eta(b_\gamma)).\] The permutations (in the vector) are of degree \(n=|G|\). Then, in Magma one can reconstruct the group as G:=PermutationGroup<n|L>, where L is the corresponding vector of lists representing permutations in imgs. The group G is a small group with identifier <n,k> [BEB, GAP21, Magma] (see the description of group).
• topreps is the list of representatives of topological equivalence classes (may not be present).
• enums is the record of enumerations where
  • epi is number of epimorphisms \(\Gamma\to G\); shown as epi in the index file
  • autg is number of \(\operatorname{Aut}(G)\)-equivalence equivalence classes; shown as autg in the index file
  • top is number of the topological equivalence classes. Since we may compute some refinement of the topological equivalence, we refer this number as ~top in the index file. However, in the case \(g=0\) or \(r=0\), the number is exact.

As concers \(g\)-admissible pairs, for genera \(17-21\) ,we provide the results of rather old experiments. Their outputs are complete at different levels. For all considered genera, the complete lists of \(g\)-admissible pairs are computed. Where possible, \(\operatorname{Aut}(G)\)-classes were determined; this is indicated by the exact number in the respective index file (otherwise the quotation mark is present in the column #actions). Re-using of data for these genera needs some extra effort. A representative (of a autg class) is given as a vector of elements of particular finitely-presented group with fixed presentation, included in the record. It may happen that the number of \(\operatorname{Aut}(G)\)-classes (and their representatives) are known though the number of epimorphisms were not set. This means that epimorphisms were computed by using LowIndexNormalSubgroups[Magma] algorithm and only the representatives of strong classes are on the output. If there are quotation marks on both, #epi and #actions, then at least one action exist, the computation was stopped (mostly due to timeout) and all computed results were committed to the output.

The census

Data last updated on Mon Aug 14 15:28:57 CEST 2023

If you have any questions concernig the data, the programs and so on, please do not hesitate to contact me.

References