On groups of PL-homeomorphisms of the real line

We investigate groups of piecewise-linear homeomorphisms of the real line; each group G is specified by the requirements that the supports of its elements f lie in a given interval I, that the slopes of f belong to a given subgroup P of the multiplicative group of positive reals and that the breakpoints of f are contained in a given P-submodule of the additive group of the reals. Our results include the determination of the orbits of G acting on I, a description of generating sets of the groups G and a study of the isomorphisms between two groups G1 = G(I1; A1, P1) and G2 = G(I2;A2,P2).