# How to construct a proper action of a group of finite virtual cohomological dimension?

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.

• Do you mean ${\mathbb Z}/2$ instead of ${\mathbb Z}/4$? – user1688 Dec 6 '15 at 17:43
• Can't you let it act on the real line with $\mathbb Z$ acting by integral translations and the generator of ${\mathbb Z}/4$ acting by multiplication by $(-1)$? – user1688 Dec 6 '15 at 17:48
• A virtually cyclic group always has proper action on the real line. – Benjamin Steinberg Dec 6 '15 at 18:13
• More generally a finitely generated group has a cocompact proper action on a tree iff it has a finite index free subgroup. – Benjamin Steinberg Dec 6 '15 at 18:15
• Yes, the point is that the group surjects $D_\infty$ with finite ($\mathbb{Z}/2$) kernel. I'm tempted to vote to close (as this was barely reserch level), but perhaps @Corbennick could at least post their comment as a answer? – HJRW Dec 6 '15 at 21:40

## 1 Answer

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).