On the Gap Conjecture concerning group growth

We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its $*$ -version) to several special classes of groups. Namely we show that its validity for the classes of simple groups and residually finite groups will imply the Gap Conjecture in full generality. A similar type reduction holds if the Conjecture is valid for residually polycyclic groups and just-infinite groups. The cases of residually solvable groups and right orderable groups are considered as well.