  We include a class of generalisations of Thompson's group $V$ introduced by
Melanie Stein into the growing framework of topological full groups. Like $V$,
Stein's groups can be described as certain piecewise linear maps with
prescribed slopes and nondifferentiable points. Stein's groups include the
class of groups where the group of slopes is generated by multiple integers
(which we call Stein's integral groups) and when the group of slopes is
generated by a single irrational number, such as Cleary's group, (which we call
irrational slope Thompson's groups). We give a unifying dynamical proof that
whenever the group of slopes is finitely generated, the simple derived subgroup
of the associated Stein's groups is finitely generated. We then study the
homology of Stein's groups, building on the work of others. This line of
inquiry allows us to compute the abelianisations and rational homology
concretely for many examples.
