New classes of topological groups defined by their continuous homomorphisms

All topological groups appearing in this talk are assumed to be Hausdorff.
Let be a class of topological groups. We say that a topological group is if every continuous homomorphism from to a group in the class is trivial. If is taken as the class of compact groups, the groups are precisely the minimally almost periodic () groups of von Neumann. classes may be compared as follows: if is a subclass of , then is contained in .
A topological group is said to satisfy the small subgroup generating property () if for every neighbourhood of the identity of there exists a family of subgroups contained in which algebraically generate a dense subgroup of .
With an operator-based approach, Dikranjan and Shakhmatov defined a hierarchy of properties (where is an ordinal) satisfying the following relationships:
(i) The property coincides with the property of Gould.
(ii) whenever .
We shall denote by the class of all topological groups which satisfy an property for some ordinal .
In this talk we present a hierarchy of classes outlined by three standard classes : locally compact, Lie and (i.e., having no small subgroup). In this new terminology, a result of Dikranjan and Shakhmatov proves that any group is . While the converse does not hold in general, we present a new result which shows that the class of Abelian groups coincides with the class Abelian groups.