Let $\Gamma$ be the semidirect product of $\mathbb{Z}$ and $\mathbb{Z}/4$,
where the action of $\mathbb{Z}/4$ on $\mathbb{Z}$ is defined by $\bar{k} \cdot x = (-1)^k x$. Clearly $\Gamma$ has virtual cohomological dimension (*vcd*) one. 


Is it possible to construct a one-dimensional contractible CW complex $X$
such that $\Gamma$ acts cellularly on $X$ and the cells have only finite
stabilizers? 


$\mathbb{Z}$ acts freely on $\mathbb{R}$ by right-shift. I tried without
success to adjust this action to obtain an action of $\Gamma$. So any help
is appreciated. 



How to construct a proper action of a group of finite virtual cohomological dimension?
You can let it act on the real line with ℤ acting by integral translations and the generator of ℤ/4 acting by multiplication by (−1).


