Small commutators of piecewise linear homeomorphisms of the real line

Abstract

IN THIS PAPER we study the group P&(R) of piecewise linear (PL) homeomorphisms of the real line R with compact support. This group has been studied by many people. As for the homology of this group, Epstein showed that the group is a perfect group and hence it is a simple group [2]. The lower-dimensional homology of this group was determined by Greenberg [S]. In particular, his result says that the 2-dimensional homology is R @Jz R and it is easy to see that the canonical bilinear map R Oz R + R is nothing but the discrete Godbillon-Vey class described in [6] and [4]. In [22], we determined all the homology group using Greenberg’s description [8] of the classifying space for transversely PL foliations. Our study on the group PL,(R) was motivated by an intention to use these results on the homology of PL,(R) to understand the homology of groups of Lipschitz homeomorphisms of R. In particular, since the (discrete) Godbillon-Vey 2-cocycle for PL,(R) was completely understood, we tried to understand the Godbillon-Vey 2-cocycle for groups of Lipschitz homeomorphisms of R using approximations by elements of the group PL,(R). The Godbillon-Vey invariant was first defined for codimension-1 C2 foliations of closed oriented 3-manifolds [7]. This invariant is the only known non-trivial invariant for C2 foliated cobordism and it varies continuously under the deformation of the foliation [ 141. For transversely oriented codimension-1 C’ foliations, the classifying space for them is contractible [ 1 S] and such foliations of closed oriented 3-manifolds are all cobordant. This fact has already been shown for transversely oriented Lipschitz foliations [l 11. Hence the Godbillon-Vey invariant cannot be defined for codimension-1 C’ or Lipschitz foliations. There are, however, a lot of classes of foliations between C’ or Lipschitz and C2. In fact, the Godbillon-Vey invariant was extended to the foliations of class C’ +a (CI > f) by Hurder and Katok [9], and to the transversely piecewise linear (or piecewise C*) by Ghys and Sergiescu [6,4]. We defined the Godbillon-Vey invariant for the foliations of class CLvyn with (1 I fl l/2) and the transversely PL foliations. Since each cobordism class of the foliations of class CL’% (1 5 j? < 2) of closed oriented 3-manifolds contains a representative which is a 2-cycle for the group GfvYn(R) of diffeomorphisms of class C ‘,% of the real line R with compact support [12], the key problem is to understand the Godbillon-Vey invariant on 2-cycles of the group G:*“(R). For a real number j3 (p 2 l), the group G, L*vn(R) of CL’Yn diffeomorphisms of R with compact support is defined as follows. An element fof G, L*Y#(R) is a Lipschitz homeomorphism with compact support such that logf’(x 0) is with bounded p-variation (see