Plane trees, cartographical groups, and Chebyshev polynomials
In this paper, plane trees are studied as a special case of graphs on a Riemann surface (or ``dessins d'enfants'' in Grothendieck's terminology). The first section is devoted to the cartographic group of such a tree \(T\); with a bipartite structure (or ``coloring'') of \(T\), the author associates several subgroups, e.g. the ``monodromy group'' of \(T\). He presents a list of 23 nonelementary irreducible trees and conjectures that it is complete. The section also contains an algorithm to compute the order of the monodromy group. In the second part, the author studies generalized Chebyshev polynomials , i.e. polynomials with at most 2 critical values, and shows that they correspond bijectively to plane trees (via the inverse image of the segment joining the critical values). For several examples, he calculates this polynomial and also the Galois orbits.
scientific article; zbMATH DE number 1213207