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Suppose a countable discrete group is acting on a smooth manifold with finite stabilizers and the orbit space is compact (and Hausdorff). How one can prove that the action is proper?

I found this statement in a paper of Luck-Rosenberg and would be happy to use it as a criterion:

  • MR2048727 Lück, Wolfgang; Rosenberg, Jonathan The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds. High-dimensional manifold topology, 322–361, World Sci. Publ., River Edge, NJ, 2003.
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    $\begingroup$ Please specify the paper by editing the question (don't give it as an "answer"). $\endgroup$ – Todd Trimble Apr 7 '15 at 10:50
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    $\begingroup$ A preprint of the Luck-Rosenberg paper is available here. On page 2, it says "Let $G$ be a (not necessarily finite) discrete group $G$. A smooth $G$-manifold $M$ is a smooth manifold with an action of $G$ by diffeomorphisms. It is called cocompact if the quotient space $G \backslash M$ is compact. It is proper if the map $G \times M \to M \times M$, $(g,m) \mapsto (g \cdot m,m)$ is proper; when the action is cocompact, this happens if and only if all isotropy groups are finite." $\endgroup$ – Dave Witte Morris Apr 7 '15 at 16:44
  • $\begingroup$ I don't think the downvote is justified. $\endgroup$ – YCor Apr 7 '15 at 18:45
  • $\begingroup$ This seems easy as long as the action is cellular. Presumably one can arrange this in the smooth setting. $\endgroup$ – HJRW Apr 8 '15 at 13:34
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An extended comment: this is not true for actions on more general locally compact (Hausdorff metrizable) spaces. For instance, let $\Gamma$ be any countable group admitting infinitely many distinct subgroups of prime order $H_n$; write $H_\infty=\{1\}$. Denote $F=\{x_m:m\le\infty\}$ (the 1-point compactification of a countable discrete set). Let $X$ be the set of pairs $(f,y)$ with $f=x_m\in F$ and $y\in \Gamma/H_m$. It is naturally a $\Gamma$-set, where $g(f,y)=(f,gy)$. Make it a topological space so that $(x_m,y)$ is isolated for $m<\infty$ and a basis of neighborhoods for $(x_\infty,g)$ with $g\in \Gamma$ is given by $(V_n(g))$, where $V_n=\{(x_m,g'H_m):m\ge n, g\in g'H_m\}$. This makes the action of $\Gamma$ continuous. It is easily seen to be compact, it's slightly more subtle that it is Hausdorff: this uses the fact that for every $g\neq 1$ in $\Gamma$, there exists $n_0$ such that $g\notin H_n$ for $n\ge n_0$. The quotient of $X$ by the $\Gamma$-action can naturally be identified to $F$ and hence is Hausdorff compact.

Clearly the stabilizers are finite. But the action is not proper: write $y_n=H_n$ consider the compact (open) subset $K$ consisting of those $(x_n,y_n)$, $n\le\infty$. Then if $s_n$ is a nontrivial element in $H_n$, $s_n$ fixes $(x_n,y_n)$, and in particular $s_nK\cap K$ is not empty, and hence the action is not compact.

At this time I'm not sure how to change this into a manifold example but I'm skeptical about the claimed result. We can start from a connected Lie group $G$, consider $G\times\mathbf{R}/\mathbf{Z}$ with the left and right $G$-actions on each leaf, mod out the leaf $G\times\{t\}$ by the action on the right of some finite subgroup $F_t$ depending smoothly on $t$ for $t\neq 0$ and tending to the trivial subgroup $F_0=\{1\}$ when $t\to 0$ (in the sense that for every compact subset $K$ of $G$ there exists $\epsilon$ such that $|t|\le\epsilon$ implies $F_t\cap K=\{1\}$; we can find such guys in $\mathbf{SL}_2(\mathbf{R})$). The issue is to arrange the resulting space (on which $G$ still acts on the left, with finite stabilizers and quotient space $\mathbf{R}/\mathbf{Z}$ to be non-singular on the slice $G\times\{0\}$ (possibly $\mathbf{R}/\mathbf{Z}$ could be replaced with another closed manifold if it helps). The next issue is to find a discrete subgroup of $G$ whose action on the resulting space remains non-proper and with Hausdorff compact quotient...

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  • $\begingroup$ PS: I tried further to get an examples following the last lines; the main issue is not smoothing (which is actually easy), nor ensuring non-properness and compact quotient, but to ensure that the quotient by the discrete subgroup is Hausdorff. $\endgroup$ – YCor Apr 7 '15 at 18:44

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