Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems

Almost nilpotency criteria and structure theorems are presented for the class of finitely generated groups of line and circle diffeomorphisms with mutually transversal elements. Key ingredients in the proof of the structure theorems are the existence/absence of an invariant measure, the (previously established) criterion for the existence of an invariant measure and restatements of this criterion in terms of various (topological, algebraic, combinatorial) characteristics of the group. The question of whether certain features of these characteristics or the existence of an invariant measure are typical for groups of line and circle diffeomorphisms is discussed. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.