# Symmetries of Surfaces, Maps and Dessins (17w5162)

Arriving in Banff, Alberta Sunday, September 24 and departing Friday September 29, 2017

## Organizers

Thomas Tucker (Colgate University)

Nigel Boston (University of Wisconsin at Madison)

Marston Conder (University of Auckland)

Gabino González-Diez (Universidad Autónoma de Madrid)

Gareth Jones (University of Southampton)

## Objectives

The main objective of this workshop is to bring together leading and emerging researchers in the area of study of actions of discrete groups on Riemann and Klein surfaces and algebraic curves, and related topics such as symmetric embeddings of graphs and dessins d'enfants on surfaces. It will follow up on a meeting held at the CIEM in Spain in 2010.

As noted in the overview statement, the last two decades have seen a burgeoning of activity, with various strands coming together, exploiting the linkages established by Belyi and Grothendieck and increasingly useful techniques from combinatorial and computational group theory. In particular, computational experiments and searches have produced a wealth of examples (either for small genera or infinite families of a particular type), and these serve as a useful test-bed for conjectures and potential new approaches.

The workshop will begin with one day of introductory lectures, presented by experts in the area, addressing recent developments and describing important open problems. Later in the week there will be a small number of shorter presentations on specific topics, but the main focus of the workshop will be on open questions. Several sessions will be devoted to these, with each consisting of a description of the problem, a summary of progress to date, a discussion of possible alternative approaches (and possibilities for future collaborations). Ideally also some of these questions will be answered, either totally or partially.

In particular, we would like participants to work on some or all of the following important open problems:

a) For large n, does chirality dominate reflexibility for orientably-regular hyperbolic maps of genus up to n? This will involve studying the growth of subgroups that are normal in the ordinary triangle group but are not necessarily normal in the full triangle group (generated by reflections).

b) What is the genus spectrum of orientable surfaces that carry a regular map with simple underlying graph? and is this spectrum full if one allows chiral maps?

c) What new techniques can be developed for obtaining the defining equations of compact Riemann surfaces given by an action of their automorphism group?

d) The symmetric genus of a group G is the smallest genus of the compact Riemann surfaces on which G has a faithful action as a group of automorphisms (which might not might not preserve orientation). Is every positive integer the symmetric genus of some group? This is a kind of 'inverse problem' which has been partially solved, but not completely. the analogous question for actions on Klein surfaces is also wide open.

e) On which subsets of regular dessins does the absolute Galois group $\Gamma$ have a faithful action? [It is known that the action is faithful on those of a given type, and on Beauville surfaces.] A good knowledge of profinite group theory techniques is essential for progress.

f) What techniques can be applied to find non-abelian actions of the absolute Galois group $\Gamma$ on specific sets of dessins? [Two examples of genus 61 are known.] In the non-abelian case, elements often imitate the Coxeter/Wilson 'hole' operators, but not always. Why is this? And how does the action of $\Gamma$ relate to that of GL(2,Z) acting on dessins as a group of hypermap operations? In many cases these two group actions commute, but again there are exceptions; are they typical? or not?

g) Maps and hypermaps are the appropriate combinatorial models of dessins, but what about in higher dimensions? For example, Beauville surfaces may be regarded as smooth quotients of products of pairs of regular hypermaps; what sort of 4-dimensional complexes describe these best?

h) Belyi's theorem yields an embedding of the absolute Galois group in the automorphism group of the profinite completion of the free group of rank 2 (the Grothendieck-Teichmüller group). But what is the nature of this embedding? i) Recent work by Gonzalez-Diez and Reyes-Carocca has shown that for an important class of complex surfaces, the so-called Kodaira fibrations (introduced by Kodaira, Atiyah and Hirzebruch as examples of fibre bundles whose signature is not multiplicative), the property of being definable over a number field depends only on the holomorphic universal cover. This contrasts strongly with Belyi's theorem, since for Riemann surfaces the holomorphic universal cover is a topological invariant. It would be very interesting to know the extent to which this phenomenon holds for arbitrary surfaces S (possibly formulated in terms of certain Zariski open sets of S).

We propose to invite participants to publish articles (possibly in a special issue of a journal) based on the work presented and progressed at the workshop. But also we propose to create a website at which the presentations and other relevant information can be stored and displayed for the benefit of the participants and others, especially younger people so they can take full advantage of their time at the workshop. This website would be made live some weeks before the workshop starts, and updated periodically afterwards.

As noted in the overview statement, the last two decades have seen a burgeoning of activity, with various strands coming together, exploiting the linkages established by Belyi and Grothendieck and increasingly useful techniques from combinatorial and computational group theory. In particular, computational experiments and searches have produced a wealth of examples (either for small genera or infinite families of a particular type), and these serve as a useful test-bed for conjectures and potential new approaches.

The workshop will begin with one day of introductory lectures, presented by experts in the area, addressing recent developments and describing important open problems. Later in the week there will be a small number of shorter presentations on specific topics, but the main focus of the workshop will be on open questions. Several sessions will be devoted to these, with each consisting of a description of the problem, a summary of progress to date, a discussion of possible alternative approaches (and possibilities for future collaborations). Ideally also some of these questions will be answered, either totally or partially.

In particular, we would like participants to work on some or all of the following important open problems:

a) For large n, does chirality dominate reflexibility for orientably-regular hyperbolic maps of genus up to n? This will involve studying the growth of subgroups that are normal in the ordinary triangle group but are not necessarily normal in the full triangle group (generated by reflections).

b) What is the genus spectrum of orientable surfaces that carry a regular map with simple underlying graph? and is this spectrum full if one allows chiral maps?

c) What new techniques can be developed for obtaining the defining equations of compact Riemann surfaces given by an action of their automorphism group?

d) The symmetric genus of a group G is the smallest genus of the compact Riemann surfaces on which G has a faithful action as a group of automorphisms (which might not might not preserve orientation). Is every positive integer the symmetric genus of some group? This is a kind of 'inverse problem' which has been partially solved, but not completely. the analogous question for actions on Klein surfaces is also wide open.

e) On which subsets of regular dessins does the absolute Galois group $\Gamma$ have a faithful action? [It is known that the action is faithful on those of a given type, and on Beauville surfaces.] A good knowledge of profinite group theory techniques is essential for progress.

f) What techniques can be applied to find non-abelian actions of the absolute Galois group $\Gamma$ on specific sets of dessins? [Two examples of genus 61 are known.] In the non-abelian case, elements often imitate the Coxeter/Wilson 'hole' operators, but not always. Why is this? And how does the action of $\Gamma$ relate to that of GL(2,Z) acting on dessins as a group of hypermap operations? In many cases these two group actions commute, but again there are exceptions; are they typical? or not?

g) Maps and hypermaps are the appropriate combinatorial models of dessins, but what about in higher dimensions? For example, Beauville surfaces may be regarded as smooth quotients of products of pairs of regular hypermaps; what sort of 4-dimensional complexes describe these best?

h) Belyi's theorem yields an embedding of the absolute Galois group in the automorphism group of the profinite completion of the free group of rank 2 (the Grothendieck-Teichmüller group). But what is the nature of this embedding? i) Recent work by Gonzalez-Diez and Reyes-Carocca has shown that for an important class of complex surfaces, the so-called Kodaira fibrations (introduced by Kodaira, Atiyah and Hirzebruch as examples of fibre bundles whose signature is not multiplicative), the property of being definable over a number field depends only on the holomorphic universal cover. This contrasts strongly with Belyi's theorem, since for Riemann surfaces the holomorphic universal cover is a topological invariant. It would be very interesting to know the extent to which this phenomenon holds for arbitrary surfaces S (possibly formulated in terms of certain Zariski open sets of S).

We propose to invite participants to publish articles (possibly in a special issue of a journal) based on the work presented and progressed at the workshop. But also we propose to create a website at which the presentations and other relevant information can be stored and displayed for the benefit of the participants and others, especially younger people so they can take full advantage of their time at the workshop. This website would be made live some weeks before the workshop starts, and updated periodically afterwards.