Small commutators of piecewise linear homeomorphisms of the real line (Q1910455)

The primary object of study is the group \(\text{PL}_c (\mathbb{R})\) of piecewise-linear homeomorphisms of \(\mathbb{R}\) with compact support. This group is shown to be dense in a certain group of Lipschitz homeomorphisms of \(\mathbb{R}\) possessing a metric. The problem addressed is whether on a subinterval \([a,b]\) of \([0,1]\) any piecewise-linear homeomorphism with support in \([a,b]\) and sufficiently close to the identity can be written as a composition of a fixed number of commutators of piecewise-linear homeomorphisms with support in \([0,1]\) and close to the identity. It is shown that the answer is yes provided that the number of non-differentiable points is bounded by a bound which depends on how far the function is from the identity.