WORKSHOP ON GALOIS-COVERS, GROTHENDIECK TEICHMÜLLER THEORY AND DESSINS D'ENFANTS
University of Leicester, Tuesday June 5-Thursday June 7, 2018
University of Leicester, Tuesday June 5-Thursday June 7, 2018
All talks and sessions take place in Ken Edwards Lecture Theatre 3. Detailed information with complete programme, titles and abstracts can be found here. A summary timetable is here. Where exist, slides of the talks can be downloaded by clicking on the titles.
Fabrizio Catanese (University of Bayreuth, Germany) Good Teichmueller spaces and holomorphic automorphisms
Abstract: One can define Teichmueller space T(X) in a quite great generality, but showing that T(X) is a complex space can be done only in special situations. The simplest one is the case of Kaehler complex structures on tori. I will first report on joint works with Corvaja, resp. Demleitner, on the case of Generalized Hyperelliptic Manifolds. Second, for surfaces X of general type, an interesting open question is whether X is rigidified (resp. : cohomologically rigidified), i.e. X does not admit automorphisms which are isotopic to the identity (resp. acting trivially on cohomology). I will briefly survey recent results by Cai-Liu-Zhang and by Liu, which show that for q >= 2, X is rigidified, and joint work with Gromadzki exhibiting surfaces of general type which are isogenous to a product and not cohomologically rigidified.
Anna Felikson (Durham University, UK) Coxeter groups, quiver mutations and hyperbolic manifolds
Abstract: Mutations of quivers were introduced by Fomin and Zelevinsky in the beginning of 2000's in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh constructed a series of presentations of finite Coxeter groups as quotients of infinite Coxeter groups. We will discuss a generalization of this construction leading to a new invariant of bordered marked surfaces, and a geometric interpretation: it occurs that presentations constructed by Barot and Marsh give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of small volume with proper actions of Coxeter groups. This work is joint with Pavel Tumarkin.
Paola Frediani (University of Pavia, Italy) On Shimura subvarieties of $A_g$ contained in the Prym locus
Abstract: I will present some results obtained in collaboration with E. Colombo, A. Ghigi and M. Penegini on Shimura subvarieties of $A_g$ generically contained in the Prym locus. I will explain the construction of 1-dimensional families of double covers compatible with a fixed group action on the base curve C such that the quotient of C by the group is the projective line. I will give a simple criterion for the image of these families under the Prym map to be a Shimura curve. I will show that this criterion allows us to construct several examples of Shimura curves generically contained in the Prym locus in $A_g$ for g<13.
Benoit Fresse (University of Lille, France) Kontsevich's graph complexes, operadic mapping spaces, and the Grothendieck-Teichmüller group
Abstract: I will report on a joint work with Victor Turchin and Thomas Willwacher about the applications of graph complexes to the study of mapping spaces associated to $E_n$-operads. The class of $E_n$-operads consists of objects that are homotopy equivalent to a reference model, the operad of little $n$-disks, which was introduced by Boardman-Vogt in topology. I will briefly review the definition of these objects. The main goal of my talk is to explain that the rational homotopy of mapping spaces of $E_n$-operads has a combinatorial description in terms of the homology of Kontsevich's graph complexes. This approach can also be used for the study of homotopy automorphism spaces associated to $E_n$-operads. In the case n=2, one can identify the result of this computation with the pro-unipotent Grothendieck-Teichmüller group. The proof of these statements relies on results on the rational homotopy of $E_n$-operads which I will also briefly explain in my talk.
Christian Gleissner (University of Bayreuth, Germany) Product Quotient Threefolds
Abstract: A complex projective variety $X=(C_1 \times ...\times C_n)/G$, where the $C_i$'s are smooth projective curves of genus at least two and $G$ is a finite group of automorphisms is called a product quotient. In the surface case $n=2$ these varieties have been studied extensively with the goal to find new surfaces of general type. However this construction is also useful to produce interesting examples of varieties of special type and in higher dimensions. In this talk we consider the threefold case. We illustrate a method to compute the plurigenera of a resolution of $X$ and apply it, in order to construct Calabi-Yau threefolds. This is joint work with R. Pignatelli and F. Favale.
Pierre Guillot (IRMA, Strasbourg, France) Introduction to dessins
Abstract: I will give an elementary overview of the theory of ''dessins d'enfants'', also known as maps on surfaces, although the term emphasizes the presence of an action of the absolute Galois group of the rational field. I will describe the action, and the relationship with the Grothendieck-Teichmüller group. I will also make some explicit computations. The talk is intended to be easy to follow, and should be a warm-up for the rest of the workshop.
Gareth Jones (University of Southampton, UK) Doubly Hurwitz Beauville Groups
Abstract: A Beauville surface is a 2-dimensional complex algebraic variety ${\mathcal S}=({\mathcal C}_1\times{\mathcal C}_2)/G$, where each ${\mathcal C}_i$ is a Belyi curve (defined over $\overline{\mathbb{Q}}$) of genus $g_i>1$, and $G$ is the automorphism group of a regular dessin on each ${\mathcal C}_i$, acting freely on ${\mathcal C}_1\times{\mathcal C}_2$. Introduced by Catanese, and named in recognition of an early example published by Beauville, these surfaces have been intensively studied by algebraic geometers such as Bauer, Catanese and Grunewald. A finite group G arises in this way, and is called a Beauville group, if and only if it has generating triples $(x_i, y_i, z_i)$ for i=1,2 with $x_iy_iz_i=1$ and no non-identity power of $x_1, y_1$ or $z_1$ conjugate to a power of $x_2, y_2$ or $z_2$. Many examples of Beauville groups are now known, including (in a deep result due to several teams of group-theorists) all non-abelian finite simple groups except $A_5$. The Hurwitz bound $|G|\le 84(g_i-1)$ implies that $\mathcal S$ has Euler characteristic $\chi({\mathcal S})\ge |G|/1764$, attained if and only if G acts on each ${\mathcal C}_i$ as a Hurwitz group, that is, each triple $(x_i,y_i,z_i)$ has type (2,3,7). Again, many examples of Hurwitz groups are known, but very few arise as Hurwitz groups in the two essentially different ways described above, as {\it doubly Hurwitz Beauville (dHB) groups}. In joint work with Emilio Pierro, we show that there are no examples among the sporadic simple groups, the small alternating groups, or various Lie-type families of small rank. However, we also show that $A_n$ is a dHB group for all $n\ge 589$ (and for some smaller n), by reinterpreting the Higman--Conder technique of `sewing coset diagrams together', used to realise $A_n$ as a Hurwitz group for sufficiently large n, as a connected sum operation on dessins; taking Galois covers then yields the required Belyi curves ${\mathcal C}_i$. Building on results about Hurwitz groups by Pellegrini and Tamburini, and by Lucchini, Tamburini and Wilson, we also show that the double cover $2.A_n$ and all groups $SL_n(q)$ and $PSL_n(q)$ are dHB groups for sufficiently large n. Joint work with Emilio Pierro.
Minhyong Kim (University of Oxford, UK) Non-abelian reciprocity laws and Diophantine geometry
Abstract: We discuss an extension of Artin's reciprocity law to general coefficients and describe its application to Diophantine geometry.
Bernhard Koeck (University of Southampton, UK) Euler characteristics and epsilon constants of wildly ramified Galois covers of curves over finite fields
Abstract: Let X be a curve over $\mathbb{F}_p$ equipped with the action of a finite group G. We consider the epsilon constants appearing in the functional equations of the corresponding Artin L-functions. Our main result relates their p-adic valuations to a certain equivariant Euler characteristic of X in the weakly ramified case. This generalises a theorem of Ted Chinburg for the tamely ramified case. It implies that these p-adic valuations are integral in the weakly ramified case. We also determine which denominators may appear in the arbitrarily wildly ramified case. This is joint work with Helena Fischbacher-Weitz and Adriano Marmora.
Michael Lönne (University of Bayreuth, Germany) Hurwitz spaces and homological stability of mapping class groups
Abstract: Given a finite group G there are Hurwitz spaces classifying algebraic curves with effective G-action or the corresponding G-covering map to the quotient curve. In previous joint work with F. Catanese and F. Perroni, stabilisation results for the number of components with specified Nielsen invariants have been proved. In terms of homological stability this starts an induction at the zero homology level. In the talk we will explain how to identify the homology groups with homology of mapping class groups with twisted coefficients and give stabilisation results obtained so far. Again this is joint work with F. Catanese and F. Perroni.
Goran Malic (University of Manchester, UK) Dessins d'Enfants and Brauer Configuration Algebras
Abstract: The monodromy data of a dessin d'enfant gives rise to a quiver whose path algebra, after imposing certain natural relations, has the structure of a symmetric special multiserial algebra. Algebras arising in such way are called Brauer Configuration Algebras (BCA) and are generalisations of the extensively studied Brauer Graph Algebras. In this talk I will give an overview of BCA's and relate them to dessins d'enfants. Of special interest will be the representation theoretic properties of BCA's that are invariant under the action of the absolute Galois group. This is joint work with Sibylle Schroll.
Marta Mazzocco (University of Birmingham, UK) Teichmueller theory for Riemann surfaces with boundaries and quantisation
Abstract: In this talk we describe a new type of surgery for non-compact Riemann surfaces that naturally appears when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincare uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmueller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmueller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesics arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as $\lambda$-lengths in Thurston--Penner terminology. We compute the Goldman bracket explicitly in terms of these $\lambda$-lengths and show that the groupoid of flip morphisms acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.
Roberto Pignatelli (University of Trento, Italy) Galois covers solving classical problems
Abstract: I will describe some new complex surfaces, Galois covers of a product of two curves, that answer two questions raised some decades ago. In the first part of the talk I will present two surfaces of general type with canonical map of very high degree, obtained in collaboration with C. Rito. By a result of A. Beauville, the degree of the canonical map of a surface of general type is at most 36, and at most 27 if the surface is irregular. Only a couple of examples are known of surfaces with canonical map of degree bigger than 16. We construct a surface with canonical map of degree 32, and an irregular surface with canonical map of degree 24. In the second part of the talk I will discuss a question posed by Kodaira and Morrow about the existence of complex manifolds that are rigid but not infinitesimally rigid. In other words, they have nontrivial infinitesimal deformations but all their small deformations are trivial. In a joint work with I. Bauer we answer affirmatively giving examples in every dimension bigger than 2. The key step is the construction of a family of Galois covers as above.
Mohamed Saïdi (University of Exeter, UK) The m-step solvable anabelian geometry of number fields
Abstract: A famous theorem of Neukirch and Uchida states that the isomorphy type of a number field is functorially encoded in the isomorphy type of its absolute Galois group. Recently, with Akio Tamagawa, we proved the following theorems. The isomorphy type of number field is determined by the isomorphy type of its maximal 3-step solvable Galois group. Further, for a positive integer m, the isomorphy type of the maximal m-step solvable extension of a number field is determined (resp. functorially) by the isomorphy type of its maximal (m+4)-step (resp. (m+5)-step) solvable Galois group. This is a substantial sharpening of the Neukirch and Uchida theorem. In my talk I will review these theorems, explain the ideas of proofs, and discuss some open questions as well as the impact these theorems might have in the future.
Leila Schneps (Institut de Mathématiques de Jussieu Paris, France) Introduction to classical and elliptic associators
Abstract: We will recall the original construction, various definitions and main properties of the Drinfeld associator, define general associators, and explain why these objects are considered to be in genus zero, due to their relation with braid groups and the moduli spaces of genus zero curves with marked points. In the second part of the talk, we will discuss how the general notion of an associator, and the Drinfeld associator in particular, can be defined in genus one.
Richard Webb (University of Cambridge, UK) How non-positively curved is the mapping class group?
Abstract: The mapping class group is the (orbifold) fundamental group of the moduli space of Riemann surfaces. In the last two decades, there has been much progress in understanding the large-scale geometry of the mapping class group and finding algebraic applications. But unfortunately many questions about the nature of its finite-index subgroups remain open. In a different setting, such as the fundamental group of a closed, hyperbolic 3-manifold, certain actions by isometries on non-positively curved cube complexes provide much information on the structure of finite-index subgroups due to the work of Agol, Wise and many others. It is therefore appropriate to ask to what extent the mapping class group admits isometric actions on non-positively curved spaces. We will show that, in general, there is no non-positively curved metric on the arc complex (a space closely related to Teichmueller space) on which the mapping class group acts by isometries---not even after restricting to finite-index subgroups.