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Groups of homeomorphisms of one-manifolds. III: Nilpotent subgroups. (English) Zbl 1037.37020
The authors consider nilpotent subgroups of $$\text{Homeo}(M)$$ and $$\text{Diff}^r(M)$$, where $$M$$ is the line $$\mathbb R$$, the circle $$S^1$$ or the interval $$I=[0,1]$$, and the paper is part of a series [Groups of homeomorphisms of one-manifolds. I: Actions of nonlinear groups, preprint (2001); Group actions on one-manifolds. II: Extensions of HĂ¶lder’s Theorem, Trans. Am. Math. Soc. 355, 4385–4396 (2003; Zbl 1025.37024)].
As J. F. Plante and W. P. Thurston [Comment. Math. Helv. 51, 567–584 (1976; Zbl 0348.57009)] discovered, $$C^2$$ regularity imposes a severe restriction on nilpotent groups of diffeomorphisms: Any nilpotent subgroup of $$\text{Diff}^2(I)$$, $$\text{Diff}^2([0,1))$$ or $$\text{Diff}^2(S^1)$$ must be abelian. The paper shows that a lowering of the regularity from $$C^2$$ to $$C^1$$ produces a sharply contrasting situation. The main results are the following.
Theorem 1. Let $$M=\mathbb R, S^1$$ or $$I$$. Then every finitely generated, torsion-free nilpotent group is isomorphic to a subgroup of $$\text{Diff}^1(M)$$.
Theorem 2. $$\text{Diff}^{\infty}(\mathbb R)$$ contains nilpotent subgroups of every degree of nilpotency.
Theorem 3. Every nilpotent subgroup of $$\text{Diff}^2(M)$$ is metabelian, i.e., has an abelian commutator subgroup.
Theorem 4. If $$N$$ is a nilpotent subgroup of $$\text{Diff}^2(M)$$ and every element of $$N$$ has a fixed-point, then $$N$$ is abelian.
Theorem 5. Let $$M=\mathbb R, S^1$$ or I. Then $$\text{Diff}_{+}^1(M)$$ contains every finitely generated, residually torsion-free nilpotent group.

##### MSC:
 37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth) 20F18 Nilpotent groups 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 37E10 Dynamical systems involving maps of the circle 57M07 Topological methods in group theory
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