On the classification of plane trees by their Galois orbit

Summary

Abstract

We consider a particular case of the Grothendieck correspondence between arithmetical and combinatorial-topological classes of objects. Our combinatorial-topological objects are plane trees (up to isotopy). The arithmetical ones are what we call the generalized Chebyshev polynomials, i.e. the polynomials in one variable with only two critical values. We consider such polynomials with complex coefficients up to affine equivalence and claim that all the equivalence classes have representatives with algebraic coefficients. The Grothendieck correspondence is established by assigning to each generalized Chebyshev polynomial its critical tree, i.e. the preimage of the segment joining its critical values; this correspondence turns out to be 1-1 in a suitable sense.

This construction allows one to define the action of the absolute Galois group Gal on the isotopy classes of plane trees. Considering this action we

1) group the isotopy classes of plane trees into some finite combinatorial classes (of bicoloured valency), called valency classes; the Galois orbits of trees are contained in these classes and often (perhaps “generally” in some sense?) coincide with them.

2) suggest some rough classification of the plane trees in terms of this action. Basically, we single out three special types of trees and call all the others general. The corresponding arithmetic turns out to be cyclotomic in two special cases and mysterious (to the author) in the one that remains.