Pro-p Trees and Applications
One of the main topics of this chapter is the study of ‘combinatorial’ constructions for pro-p groups. We are interested in those groups that can be defined by means of universal properties, such as free pro-p groups, free and amalgamated free products, HNN-extensions, etc. In some sense these are the basic building blocks of pro-p groups. As in the case of abstract groups, such constructions and their properties can be often best understood by studying the action of the group arising from the constructions on a natural ‘tree’ determined by the construction. The Bass—Serre theory of abstract groups acting on trees gives a complete and satisfying description of those groups as fundamental groups of certain graphs of groups. There is a concept of ‘tree’ which is appropriate for the study of pro-p groups, namely the so-called pro-p tree; we define it here and develop its main properties. By contrast with the situation for abstract groups acting on trees, the structure of a pro-p group acting on a pro-p tree is not yet fully clear. But this is one of the reasons that make this area so attractive, for it is still full of open and interesting problems. Yet, one has a considerable number of results about pro-p groups acting on a pro-p trees, and we give an ample sample of them in the first four sections of this chapter.
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