Some questions about the dimension of a group action

Abstract

We discuss three families of groups, $ZW_n$, $PL(I^n)$, and $PL(S^n)$ (the last two being families of groups of piecewise-linear homeomorphisms of standard $n$-dimensional spaces). We note that for all positive integers $n$, $ZW_n$ embeds in $PL(I^n)$ which embeds in $PL(S^n)$. In another direction, $ZW_2$ fails to embed in $PL(I^1)$ by an earlier result of the author, and we extend that result to show that $ZW_2$ also fails to embed in $PL(S^1)$. The nature of the proofs of these non-embedding results leads us to ask if there are corresponding non-embedding results in higher dimensions.