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Register a new userDessins dEnfants for Single-Cycle Belyi Maps
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Riemanns Existence Theorem gives the following bijections: (1) Isomorphism classes of Belyi maps of degree $d$. (2) Equivalence classes of generating systems of degree $d$. (3) Isomorphism classes of dessins denfants with $d$ edges. In previous work, the first author and collaborators exploited the correspondence between Belyi maps and their generating systems to provide explicit equations for two infinite families of dynamical Belyi maps. We complete this picture by describing the dessins denfants for these two families.
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We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on the rational dynamics.
Almost two decades ago Zapponi introduced a notion of orientability of a clean dessin denfant, based on an orientation of the embedded bipartite graph. We extend this concept, which we call Z-orientability to distinguish it from the traditional topological definition, to the wider context of all dessins, and we use it to define a concept of twist orientability, which also takes account of the Z-orientability properties of those dessins obtained by permuting the roles of white and black vertices and face-centres. We observe that these properties are Galois-invariant, and we study the extent to which they are determined by the standard invariants such as the passport and the monodromy and automorphism groups. We find that in general they are independent of these invariants, but in the case of regular dessins they are determined by the monodromy group.
It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin denfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compact dessins. Moreover, we show that any tame action of a countable group is so realizable.
We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of their Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. This allows us to study the behavior of the monodromy groups under specialization of the maps, and to derive applications to dynamical sequences.
We prove existence and nonexistence results for certain differential forms in positive characteristic, called {em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$.
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