Rotislav Grigorchuk: On the Gap Conjecture for group growth and related topics

I will give a survey of results around the Gap Conjecture which was formulated by the speaker in 1989 and which states that if the growth of a finitely generated group is less than e^{\sqrt n}, then it is polynomial. If proved, it will be a far reaching improvement of Gromov's remarkable theorem describing groups of polynomial growth as virtually nilpotent groups. Old and new results around the conjecture and its weaker and stronger forms will be formulated and discussed.