On the reduction of GAP Conjecture to certain classes of groups and around

In this note we revise our approach from [Rostislav Grigorchuk. On the gap conjecture concerning group growth. Bull. Math. Sci., 4(1):113--128, 2014] to the reduction of the Gap Conjecture on Group Growth to the class of just-infinite groups. For this purpose we introduce the class of groups that we call (GAPSY) as well as some other classes. The properties of these classes are discussed in the relation with growth, elementary amenability, and just-infiniteness. Additionally, we discuss the GAP type problems for cancellative semigroups. A number of (old and new) open problems and few historical remarks is included in the note as well.