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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.

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  • $\begingroup$ Do you mean ${\mathbb Z}/2$ instead of ${\mathbb Z}/4$? $\endgroup$ – user1688 Dec 6 '15 at 17:43
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    $\begingroup$ 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 $(-1)$? $\endgroup$ – user1688 Dec 6 '15 at 17:48
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    $\begingroup$ A virtually cyclic group always has proper action on the real line. $\endgroup$ – Benjamin Steinberg Dec 6 '15 at 18:13
  • $\begingroup$ More generally a finitely generated group has a cocompact proper action on a tree iff it has a finite index free subgroup. $\endgroup$ – Benjamin Steinberg Dec 6 '15 at 18:15
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    $\begingroup$ 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? $\endgroup$ – HJRW Dec 6 '15 at 21:40
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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).

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