An algebraic classification of some solvable groups of homeomorphisms. (Q2482063)

The author gives two different algebraic descriptions of the isomorphism classes of solvable subgroups of the group \(\text{PL}_0(I)\) of piecewise-linear orientation-preserving homeomorphisms of the unit interval \(I\), and also of the generalized R.~Thompson groups \(F_n\). The first description is as a set of isomorphism classes of groups which is closed under three algebraic operations. The second is as a set of isomorphism classes of subgroups of a countable collection of groups obtained inductively by a wreath product construction. The main result is that these two descriptions give the same class of groups. The methods are a careful study of the dynamics of groups of piecewise linear homeomorphisms with respect to the ``orbitals'' of single elements \(h\in H\), where an orbital is a connected component of the complement \(\{x\in I;\;h(x)\neq x\}\) of the support of \(h\). The paper uses the geometric results of the author's paper [J. Lond. Math. Soc., II. Ser. 78, No. 2, 352-372 (2008; Zbl 1192.20025)].