Schedule for: 17w5162 - Symmetries of Surfaces, Maps and Dessins

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Introduction (by Marston Conder and Tom Tucker) to regular maps and edge-transitive maps, including some important open questions in this theme area

Introduction (by Ernesto Girondo and Gabino González-Diez) to Belyi theory and the absolute Galois group, including some important open questions in this theme area

Introduction (by Allen Broughton and Aaron Wootton) to defining equations for a Riemann surface, including some important open questions in this theme area

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Additional themes, including some important open questions in each theme area: (a) Theme 4: Properties of higher rank structures (introduced by Dimitri Leemans); (b) Theme 6: $n$-gonal morphisms of Riemann surfaces (introduced by Mariela Carvacho); (c) Theme 5: Cayley maps, skew morphisms and Hamiltonicity of regular maps (introduced by Robert Jajcay).

In the late 1990s, Thomas Breuer created an algorithm, and wrote corresponding computer code, which determines all groups acting on Riemann surfaces of a fixed genus. His code also determined generating vectors for the actions, but he did not record that data. The results of his program up to genus 15, along with additional information such as if the Riemann surface is hyperelliptic or cyclic trigonal, the decomposition of the corresponding Jacobian Variety, and the dimension of the corresponding Shimura variety, can now be found online at: http://www.lmfdb.org/HigherGenus/C/Aut/. We’ll briefly introduce these pages, discuss features of them, and mention future planned additions.

I will present a number of relevant results which may be not known in the ribbon graph or dessins communities

In this talk we will briefly summarise what is known about almost simple groups acting regularly/chirally on abstract polytopes, and state some open questions

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

An introduction to Theme 7 (on uper-symmetries of regular maps)

An introduction to Theme 8 (other aspects of group actions on surfaces, such as pseudo-real surfaces and purely non-free actions) by Grzegorz Gromadzki and Javier Cirre, including some important open questions in this theme area

Working groups on the first four main themes: 1) Regular and Edge-Transitive Maps 2) Belyi theory & the absolute Galois group 3) Defining equations for a Riemann surfaces 4) Properties of higher rank structures.

During numerous talks and conferences, a question was often raised: How to find a complete set of combinatorial and/or group-theoretic invariants of the Galois action on dessins. We will show that this is impossible. Indeed, sometimes combinatorial orbits of dessins split into several Galois orbits because of some subtle Diophantine relations between degrees of of vertices and faces.

A general problem is to understand all (injective) homomorphisms between (finite index subgroups of) mapping class groups of surfaces. Birman and Hilden proved that if $S\rightarrow X$ is a regular branched covering space of surfaces, there is an embedding of the subgroup of the mapping class group of $X$ consisting of mapping classes that have representatives that lift to $S$ in the mapping class group of $S$ modulo the group of deck transformations. This relationship does not always hold for irregular branched covers. We give a necessary condition and a sufficient condition for when such an embedding exists. We also give new explicit examples that satisfy the necessary condition and examples that do not satisfy the sufficient condition.

In this talk we introduce a surprising correspondence between $(m,n)$-complete regular dessins and admissible pairs of skew-morphisms of the cyclic groups of orders $m$ and $n$. A skew-morphism $\varphi$ of a finite group $A$ is a permutation on $A$ such that $\varphi(1)=1$ and $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$ where $\pi:A\to\mathbb{Z}_{|\varphi|}$ is an integer function. We determine the pairs $(m,n)$ for which there exists exactly one dual pair of $(m,n)$-complete regular dessins, thus generalising an earlier result by Jones, Nedela and \v Skoviera (2008). This is joint work with Y.Q.Feng, Kan Hu and M. {\v S}koviera.

Define QC$(n)$ to be the number of isomorphisms classes of quasiplatonic cyclic $n$-gonal surfaces. We use formulas of R.\ Benim and A.\ Wootton to describe a method for computing QC$(n)$. We also relate QC$(n)$ to the number of regular dessins d'enfants having a cyclic group of automorphisms of order $n$.

Working groups on the next four main themes: 5) Cayley maps, skew morphisms and Hamiltonicity of regular maps 6) n-gonal morphisms of Riemann surfaces 7) "Super-symmetries" of regular maps (duality, etc.) 8) Other aspects of group actions on surfaces, such as pseudo-real surfaces and purely non-free actions.

A (face-)primer hypermap is a regular oriented hypermap whose orientation-preserving automorphism group $G$ acts faithfully on the hyperfaces. In this paper, we investigate the primer hypermaps for which $G$ is nilpotent and the hypervertex-valency is a prime $p$. (We call these PNp-hypermaps.) We prove that for any PNp-hypermap, the group $G$ must be a finite $p$-group, and the number of hyperfaces is bounded above by a function of the nilpotency class of $G$. Moreover, we show that for any positive integer $c$, there is a unique PNp hypermap $H_c$ of class $c$ attaining the given bound, and every other PNp hypermap of class at most $c$ is a quotient of $H_c$.

Except very few cases (7 in total), if a regular dessin of genus $g$ has rotational group of order $4g$, then the dessin lies on Wiman's curve with $8g$ automorphisms.

Teichmüller curves are totally geodesic complex curves inside the moduli space of Riemann surfaces. Under a Prym-Torelli map, one can see them as curves in some Hilbert modular variety, which parametrises abelian varieties with real multiplication. We give an overview of how to cut out these curves as the vanishing locus of Hilbert modular forms and the invariants that can be computed from this viewpoint.

Cost is CAD 30 per person (or CAD35 if feeling generous) for those who indicated they would take part. Free afternoon for those who indicated they would NOT take part.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.