  We study algebraic properties of groups of PL or smooth homeomorphisms of
unit cubes in any dimension, fixed pointwise on the boundary, and more
generally PL or smooth groups acting on manifolds and fixing pointwise a
submanifold of codimension 1 (resp. codimension 2), and show that such groups
are locally indicable (resp. circularly orderable). We also give many examples
of interesting groups that can act, and discuss some other algebraic
constraints that such groups must satisfy, including the fact that a group of
PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no
elements that are more than exponentially distorted.
