Some questions about the dimension of a group action

This article discusses three families of groups: Z≀(Z^n), PL(I^n), and PL(S^n) (the last two being the families of groups of piecewise-linear homeomorphisms of standard n-dimensional spaces). It is shown that for positive n ∈ N, Z≀(Z^n) embeds in PL(I^n), which embeds in PL(S^n). It is known that Z≀(Z^2) fails to embed in PL(I^1), and this article extends that previous result to show that Z≀(Z^2) also fails to embed in PL(S^1). The nature of the proofs of these embedding and non-embedding results hints that there may be corresponding non-embedding results in higher dimensions.