Near actions

Abstract

A near permutation of a set is a bijection between two cofinite subsets, modulo coincidence on smaller cofinite subsets. Near permutations of a set form its near symmetric group. In this monograph, we define near actions as homomorphisms into this group, and perform a systematic study of this notion. Notably, we deal with the problem of realizability: how far is a near action from being induced by an action? Also, what invariants of group actions are actually invariants of the underlying near action? the most fundamental one is the space of ends. We notably provide a classification of near actions of finitely generated abelian groups. There are several examples of exotic near actions of the free abelian group of rank two, which we classify according to a positive integer, called winding number, and an integral vector, called additive holonomy. In contrast, we prove that there are no such exotic near actions in higher rank abelian groups. We also obtain results for other groups; for instance, every near action of SL(2,Z) is realizable; this is not true for arbitrary amalgams of two finite cyclic groups. We introduce and study almost and near automorphism groups, which have been considered for some particular graphs or labeled graphs, in the general setting of relational structures.