17 research outputs found
Hypermap operations of finite order
Duality and chirality are examples of operations of order 2 on hypermaps.
James showed that the groups of all operations on hypermaps and on oriented
hypermaps can be identified with the outer automorphism groups Out ∼=
PGL2(Z) and Out + ∼=
GL2(Z) of the groups = C2 ∗C2 ∗C2 and + = F2. We
will consider the elements of finite order in these two groups, and the operations
they induce
Unified bijections for planar hypermaps with general cycle-length constraints
We present a general bijective approach to planar hypermaps with two main
results. First we obtain unified bijections for all classes of maps or
hypermaps defined by face-degree constraints and girth constraints. To any such
class we associate bijectively a class of plane trees characterized by local
constraints. This unifies and greatly generalizes several bijections for maps
and hypermaps. Second, we present yet another level of generalization of the
bijective approach by considering classes of maps with non-uniform girth
constraints. More precisely, we consider "well-charged maps", which are maps
with an assignment of "charges" (real numbers) on vertices and faces, with the
constraints that the length of any cycle of the map is at least equal to the
sum of the charges of the vertices and faces enclosed by the cycle. We obtain a
bijection between charged hypermaps and a class of plane trees characterized by
local constraints
Maps on surfaces and Galois groups
A brief survey of some of the connections between maps on surfaces, permutations, Riemann surfaces, algebraic curves and Galois groups is given
Hypercubes as dessins d'enfant
We describe the action of the group GL2(Z) on embeddings of hypercubes on compact
orientable surfaces, specifically classifying the elements of finite order that can change
the genus of the underlying surface by an arbitrarily large amount. In doing so we give
an explicit illustration of the kind of computations encountered in the study of dessins
d'enfants in the hope that those new to the area may find such an explicit example useful