Groups of homeomorphisms of one-manifolds. III: Nilpotent subgroups.

*(English)*Zbl 1037.37020The 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.

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.

Reviewer: Eugene Ershov (St. Peterburg)

##### 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 |