Geodesic
Groups of line and circle homeomorphisms. Criteria for almost nilpotency
Sbornik. Mathematics, Tome 210 (2019) no. 4, pp. 495-507

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For finitely-generated groups of line and circle homeomorphisms a criterion for their being almost nilpotent is established in terms of free two-generator subsemigroups and the condition of maximality. Previously the author found a criterion for almost nilpotency stated in terms of free two-generator subsemigroups for finitely generated groups of line and circle homeomorphisms that are $C^{(1)}$-smooth and mutually transversal. In addition, for groups of diffeomorphisms, structure theorems were established and a number of characteristics of such groups were proved to be typical. It was also shown that, in the space of finitely generated groups of $C^{(1)}$-diffeomorphisms with a prescribed number of generators, the set of groups with mutually transversal elements contains a countable intersection of open dense subsets (is residual). Navas has also obtained a criterion for the almost nilpotency of groups of $C^{(1+\alpha)}$-diffeomorphisms of an interval, where $\alpha>0$, in terms of free subsemigroups on two generators. Bibliography: 21 titles.
Keywords: almost nilpotency, group of line or circle homeomorphisms, free subsemigroup. 
L. A. Beklaryan. Groups of line and circle homeomorphisms. Criteria for almost nilpotency. Sbornik. Mathematics, Tome 210 (2019) no. 4, pp. 495-507. http://geodesic.mathdoc.fr/item/SM_2019_210_4_a1/
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