Embeddings of trees in the hyperbolic disk

Abstract

We consider the conditions under which a homogeneous tree of even degree (i.e. the Cayley graph of a free group) may be embedded in the hyperbolic disk in such a way that the automorphisms induced by rotation and by translation by group elements may be represented by automorphisms of the disk. We concentrate on the study of a particular optimal such embedding. We show that in this case bounded analytic functions on the disk are determined by their values at the vertices of the embedded tree. Using a construction of we can define harmonicity on a homogeneous tree in such a way that bounded functions on the disk restrict to harmonic functions on the tree if and only if they are harmonic. We also show a simple and elegant construction of this tree.

AMS No:

• 30F35
• 05C05
• 30E05
• 31C05