Let $\Gamma$ be a discrete countable infinite Group and let X be compact minimal $\Gamma$-space. A $\Gamma$-compactification by X is a compact topology on $\Gamma\cup X$ on which $\Gamma$ acts continuously by left multiplication and the original action on X respectively, and such that $\Gamma$ is dense in $\Gamma\cup$ X.

Is there more than one way to "glue" X to $\Gamma$ in such a way? Are there canonical families of $\Gamma$ compactifications? and what all of this has to do with the old and famous Borsuk's non-retract theorem from classical topology?