We construct finitely generated groups of small period growth, i.e. groups where the maximum order of an element of word length n grows very slowly in n. This answers a question of Bradford related to the lawlessness growth of groups and is connected to an approximative version of the restricted Burnside problem.

Intersection growth concerns the asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this paper we show that the intersection growth of some groups may not be a nicely behaved function by showing the following seemingly contradictory results: (a) for any group G the intersection growth function iG(n) is super linear infinitely often, and (b) for any non-decreasing unbounded function f there exists a group G such that the graph of iG is below the one of f infinitely often.

The following theorem is proved. Let m, k and n be positive integers. There exists a number η=η(m,k,n) depending only on m, k and n such that if G is any residually finite group satisfying the condition that the product of any η commutators of the form [xm,y1,…,yk ] is of order dividing n, then the verbal subgroup of G corresponding to the word w=[xm,y1,…,yk ] is locally finite.

We define equivariant completion of a G-complex and define residually finite G-spaces. We show that the group of G-homotopy classes of G-homotopy self equivalences of a finite, residually finite G-complex, in residually finite. This generalizes some results of Roitberg.