David Roberts cocompact proper Lie groupoids

Some notes on the structure theory of cocompact proper Lie groupoids. All Lie groupoids will be comprised of Hausdorff paracompact smooth manifolds.

Definition

A Lie groupoid $X$ is called proper if the map $(s,t)\colon X_1 \to X_0\times X_0$ is proper.

Proposition

Given a proper Lie groupoid the topological space $X_0/X_1$, the coequaliser of $X_1\rightrightarrows X_0$, is Hausdorff and paracompact. In addition, the map $X_0 \to X_0/X_1$ is open.

Proposition

The orbits of a proper Lie groupoid $X$ are closed embedded submanifolds of $X_0$, and all isotropy groups are compact Lie groups.

Definition

A proper Lie groupoid is called cocompact if $X_0/X_1$ is compact.

Recall the definition of weak equivalence of Lie groupoids: an internal functor $X\to Y$ that is fully faithful and such that $t\circ pr_2\colon X_0 \times_{Y_0,s} Y_1 \to Y_0$ is a surjective submersion. We shall be interested in finding, for a given cocompact proper Lie groupoid, another Lie groupoid connected by a span of weak equivalences to the first and having a particular form. This requires the following simple result.

Lemma

The properties of being proper and cocompact are invariant under weak equivalence.

The main technical result we shall use is the slice theorem for proper Lie groupoids, detailed expositions of which can be found in Crainic and Struchiner and Pflaum et al. To state the theorem we need to discuss the linearisation of a proper Lie groupoid.

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Proposition (Pflaum et al)

For a proper Lie groupoid $X$, the space $X_0/X_1$ admits arbitrarily fine good open covers.

Thus the orbit space of a cocompact proper Lie groupoid admits arbitrarily fine finite good open covers.

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Proposition

A cocompact proper Lie groupoid $X$ admits an essentially surjective functor from a Lie groupoid of the form

$$\coprod_{i=1}^N B^{n_i}//G_i$$

where $G_i$ is a compact Lie group and $B^{n_i}$ is an invariant open ball in an orthogonal representation of $G_i$, and each $B^{n_i}//G_i \hookrightarrow X$ is a full embedding.

In fact any proper Lie groupoid admits such a functor where the coproduct isn’t necessarily finite, and this is essentially the content of the linearisation theorem. Compactness of the orbit space together with openness of the quotient map means finitely many balls map to an open cover of the orbit space, hence the saturations of those open balls form an open cover of the object manifold of the groupoid.

Definition

Call a Lie groupoid of the form $B^n//G$ as in the previous proposition a ball groupoid.

Corollary

A cocompact proper Lie groupoid has only finitely many normal orbit types, and hence only finitely many weak orbit types.

The weak orbit type of a object $p$ is the isomorphism class of the Lie group of automorphisms of $p$. The normal orbit type of $p$ is the isomorphism class of the slice representation at $p$ (see Pflaum et al, definition 5.6).

Given two ball groupoids $B^{n_1}//G_1$, $B^{n_2}//G_2$ mapping to a Lie groupoid $X$, it is of great interest to know what the pullback $B^{n_1}//G_1\times_X B^{n_2}//G_2$ (in the appropriate Lie groupoid sense) is. One immediate result is that if, without loss of generality, $n_1 \leq n_2$, then it can be taken to be a full open subgroupoid of $B^{n_1}//G_1$. Since this ball groupoid is the interior of the action groupoid $\overline{B}^{n_1}//G_1$, which is a cocompact proper Lie groupoid, it can itself (I believe!) be covered by finitely many ball groupoids, in the sense of the above proposition.

However, ideally the subgroupoid $\Gamma \subset B^{n_1}//G_1$ has as object space a contractible open set in $B^{n_1}$. One way to get at this is to get estimates on a sort of ‘transverse convexity radius’. This will ultimately depend on the following result.

Theorem

Every proper Lie groupoid $X$ admits a metric on $X_0$ that is transversally invariant and complete, and that is adapted to the singular Riemannian foliation given by connected components of the orbits of $X$

A key property of a singular Riemannian foliation is that every geodesic that is perpendicular to a leaf at one point is perpendicular to every leaf it meets.

What if I consider the convexity radius for this metric?

We also can consider the injectivity radius (or a transverse version), which will tell us how large the ball groupoids can be. Actually the local structure theorem is a little stronger, in that one gets charts $U\ni x$ in the object manifold such that the induced groupoid $X[U]$ is isomorphic to $codisc(D)\times (B^n//G)$. Presumably if we take the geodesic radius of $U$ to be such that it arises from via exponential map and is geodesically convex, then since the decomposition $D\times B^n$ is in some sense compatible with the geodesic flow, we should get $B^n$ transversally geodesically convex. Moreover, for any point $p\in U$ the hope is that there is a global slice through $p$ for $X[V]$ for any neighbourhood $V$ of $p$ such that $V\cap \mathcal{O}_x = \emptyset$ (here $\mathcal{O}$ is the orbit through $x$).

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Conjecture

Cocompact proper Lie groupoids are compact objects in the (2,1)-category of Lie groupoids and anafunctors (equivalently the (2,1)-category of differentiable stacks).

This is complicated somewhat by differing notions of what ‘compact’ means in a higher topos. See differential cohomology in a cohesive topos, section 3.6.4.

References

• M. Crainic, I. Struchiner, On the linearization theorem for proper Lie groupoids arXiv:1103.5245

• M.J. Pflaum, H. Posthuma, X. Tang, Geometry of orbit spaces of proper Lie groupoids, arXiv:1101.0180