From a purely algebraic point of view, there is not a lot one can say about infinite groups in general. Traditionally, these have been studied to good effect in combination with topology or geometry. These lectures represent an introduction to some recent developments that arise out of looking at infinite groups from a point of view inspired – in a general sense – by number theory; specifically the interaction between ‘local’ and ‘global’, where by ‘local’ properties of a group G, in this context, one means the properties of its finite quotients, or equivalently properties of its profinite completion Ĝ. The second chapter directly addresses the interplay between certain finitely generated groups and their finite images. The other two chapters are more specifically ‘local’ in emphasis: Chapter I concerns the algebraic structure of certain pro-p groups, while Chapter III introduces a way of studying the rich arithmetical data encoded in certain infinite groups and related structures.

A motivating example for all of the above is the question of ‘subgroup growth’. Say G has sn(G) subgroups of index at most n for each n; the function n ↦ sn(G) is the subgroup growth function of G, and is finite-valued if we assume that G is finitely generated. Now we can ask (inspired perhaps by Gromov's celebrated polynomial growth theorem): what does it mean for the global structure of a finitely generated group if its subgroup growth function is (bounded by a) polynomial?