Some questions about the dimension of a group action

Collin Bleak

doi:10.1112/blms/bdn060

paper

Some questions about the dimension of a group action

arXiv:math/0611434 · doi:10.1112/blms/bdn060

Abstract

We discuss three families of groups, $Z W_{n}$ , $P L (I^{n})$ , and $P L (S^{n})$ (the last two being families of groups of piecewise-linear homeomorphisms of standard $n$ -dimensional spaces). We note that for all positive integers $n$ , $Z W_{n}$ embeds in $P L (I^{n})$ which embeds in $P L (S^{n})$ . In another direction, $Z W_{2}$ fails to embed in $P L (I^{1})$ by an earlier result of the author, and we extend that result to show that $Z W_{2}$ also fails to embed in $P L (S^{1})$ . The nature of the proofs of these non-embedding results leads us to ask if there are corresponding non-embedding results in higher dimensions.

8 Pages, follows up on some loose ideas in "A geometric classification of some solvable groups of homeomorphisms." Arxiv: math.GR/0602036