For this question, I am in the smooth finite-dimensional category. All objects are $\mathcal C^\infty$ manifolds and all maps are smooth. Recall the notion of [groupoid](http://www.google.com/search?hl=&q=groupoid), and let's restrict our attention to those for which the source map (and hence also the target map) is a (necessarily surjective) submersion. If $G\rightrightarrows X$ is a groupoid, me define a **subgroupoid over $X$** to be an immersed submanifold $H\hookrightarrow G$ so that $H\rightrightarrows X$ is a groupoid. (If this is not the standard definition, let me know.)


Pick $x\in X$. Does there exist a subgroupoid $H\hookrightarrow G$ so that: (1) $x$ has no automorphisms in $H$, and (2) if $y\in X$ with $x\cong y$ in $G$, then $x\cong y$ in $H$? I.e.: the groupoid $H$ should be as small as possible but still preserve the isomorphism class of $x$. I don't mind if other isomorphism classes shrink in size.



Given an object in a Lie groupoid, does there exist a subgroupoid for which the object has no automorphisms but retains its equivalence class?
Consider the case of a group $K$ acting on $X$. Restricting to the orbit of the point $x$, by the orbit-stabilizer theorem $X$ is identified with $K/\text{Stab}(x)$. Your question seems to amount to asking whether the projection $K\times K/\text{Stab}(x)\to K/\text{Stab}(x)\times K/\text{Stab}(x)$ given by $(\gamma,p)\mapsto (\gamma\cdot p,p)$ admits an immersed section. Restricting such a section to a fiber $K/\text{Stab}(x)\times \text{pt}$ would give an section of the map $K\to K/\text{Stab}(x)$, which will not exist in general. (Edit: rewritten. Thanks to David Carchedi for pointing out the mistake in my notation.)


A concrete counterexample: consider the action of $K=\mathbb{R}$ on $X=S^1$ by translation. Then $G$ is diffeomorphic to $\mathbb{R}\times S^1$, and $H$ would have to be diffeomorphic to $S^1\times S^1$, but $S^1\times S^1$ does not immerse in $\mathbb{R}\times S^1$. Moreover the restriction to a fiber $S^1\times \text{pt}$ would be an immersion of $S^1$ into $\mathbb{R}$, which also cannot exist.


But this is not a phenomenon just of fundamental group or of equi-dimensional immersions. For example, consider $K=\text{Isom}(\mathbb{H}^2)=\text{PSL}\_2\mathbb{R}$ acting on a hyperbolic surface $\Sigma$. A section of $\text{PSL}\_2\mathbb{R}\times \Sigma\to \Sigma\times \Sigma$ restricts to an immersed section of $\text{PSL}\_2\mathbb{R}\to \Sigma$. Of course $\Sigma$ *does* immerse, in fact embed, in $\text{PSL}\_2\mathbb{R}$ (it's a 3-manifold). But if we had a continuous section $\varphi:\Sigma\to \text{PSL}\_2\mathbb{R}$, we could translate a nonzero vector $v$ to each point $p$ by the isometry $\varphi(p)$. This would yield a nonzero vector field on $\Sigma$, contradicting the Gauss-Bonnet theorem. The same works for $\text{SO}(3)$ acting on $S^2$ or $\mathbb{R}P^2$, where the fundamental group is not the issue.


