Theory of groups that act on trees

Abstract

In this note we present a comparatively short combinatorial proof of the theorem of Bass and Serre [i] on the structure of a group that acts on a tree by automorphisms. In contrast to the known proofs of this theorem [i-3], based on the ideas of the theory of covering spaces or on suitable topological interpretations of groups, we use here the method of fundamental regions in the form presented by Macbeath [4], with a consequent simplification of the defining relations by Tits transformations. Such an approach leads naturally to the problem of a combinatorial proof of Macbeath's theorem in the case of the action of a group on a tree; the corresponding proof, together with all the necessary definitions, is contained in Sec. i. The theorem of Bass and Serre itself is proved in Sec. 2, where we give some well-known corollaries of this theorem.