Ends and algebraic directions of pseudogroups

Summary

Abstract

The notion of an end of a group is generalised to the case of finitely generated pseudogroups. The notion of a direction of a pseudogroup, subtler than the end of a pseudogroup, is introduced in this paper. The ends and directions of the Markov pseudogroup are described.

Introduction

The notion of an end of a group was originated in the 40's by H. Freudenthal [Fre] and H. Hopf [Hop]. A classical result of H. Hopf [Hop] says that the end space E(X) of a connected topological space X is an invariant of the group G of covering transformations. Thus it becomes meaningful to define the end space of the finitely generated group G as E(G) ≔ E(X).

The notion of the end of a group was studied again by J. Stailings [Sta] and D. Cohen [Cohl], [Coh2] in the 70's. D. Cohen has used a combinatorial method to define the number of ends of an arbitrary group G. The end spaces of finitely generated groups were completely classified (see [Coh2], [Hop], [Sta]).

The theory of ends of groups was generalised. A more general concept, introduced by C. Hougton [Hou] and further explored by G. Scott [Sco] and M. Sageev [Sag], is the number of a group G relative to a subgroup H.