Problems on groupoids

Some questions on groupoids

Additions, corrections and comments should be sent to Jean Renault, renault@labomath.univ-orleans.fr or directly to the contributor.

Contributor: André Haefliger
June 1999, Boulder Question: Let $\cal F$ be a foliation on a manifold M such that the universal covering of each leaf is contractible and T be a transveral. Let G be the fundamental groupoid of $\cal F$ associated with T. Prove that for every G-sheaf $\cal A$ on T, the map $\begin{displaymath}H^*(G,{\cal A})\rightarrow H^*(M,{\cal A})\end{displaymath}$
is an isomorphism.
Answer proposed by Ieke Moerdijk at January 22, 2001.

Contributor: Steve Hurder
June 1999, Boulder Question: Let $\cal F$ be an amenable C¹ foliation on a compact manifold M.
a) If L is a compact leaf, must its holonomy group be amenable?
b) Same question for a dense leaf L.

Contributor: Jerry Kaminker
June 1999, Boulder Preamble. Let G be a Lie groupoid and $\cal G$ its Lie algebroid. Then we have the pseudo-differential operators extension $\begin{displaymath}0\rightarrow C^*(G)\rightarrow \Psi^0(G)\rightarrow C_0(S{\cal G}^*)\rightarrow 0 \leqno{(1)}\end{displaymath}$
as an element of $KK^1(C_0(S{\cal G}^*), C^*(G))$ .
In the case when $G=M\times M$ , the extension
$\begin{displaymath}0\rightarrow {\cal K}\rightarrow \Psi^0(G)\rightarrow C_0(S^*M)\rightarrow 0\end{displaymath}$
is determined by the symplectic structure of S^*M via the Index Theorem for Families.
Question: What does the Poisson algebra structure of $C^\infty({\cal G}^*)$ determine about (1) ?
Does it determine the cohomology class needed in a topological index formula?

Contributor: Alex Kumjian
June 1999, Boulder Question: Find a groupoid cohomology (with coefficients) valid for locally compact groupoids such that the following conditions are satisfied:
1) Theory agrees with Grothendieck's equivariant sheaf cohomology if the groupoid is étale.
2) Theory agrees with Moore's cohomology if the groupoid is a locally compact group.
3) An equivalence of groupoids induces an isomorphism of cohomology.
4) The second cohomology with the circle as coefficients may be identified with the Brauer group of the groupoid.

Contributor: Alex Kumjian
June 1999, Boulder Question: Let $\phi:\Gamma\rightarrow\Lambda$ be a continuous open morphism (surjective) between groupoids. Under what conditions does one get a Fell bundle E over $\Lambda$ such that $E_\lambda$ is the closure of $C_c(\phi^{-1}(\lambda))$ (under an appropriate norm) and $\begin{displaymath}C^*(\Gamma)\simeq C^*(\Lambda,E) ?\end{displaymath}$

Contributor: Paul Muhly
June 1999, Boulder Question: Let G be a principal, étale groupoid. When does there exist a ``faithful'' 1-cocycle $c:G\rightarrow {\bf R}$ , i.e. a continuous homomorphism $c:G\rightarrow {\bf R}$ such that c^-1(0)=G⁽⁰⁾ ? A necessary condition is amenability, but is it sufficient?

Contributor: Chris Phillips
June 1999, Boulder Question: Let G be an amenable étale essentially principal groupoid. When is C_r^*(G) a stably finite C^*-algebra?

Contributor: Chris Phillips
June 1999, Boulder Question: Let G be an étale groupoid and let X be a proper G-space. Can K⁰_G(X) be described using finite dimensional vector bundles only?

Contributor: Birant Ramazan
June 1999, Boulder Preamble. Suppose that $\cal G$ is the Lie algebroid of a Lie groupoid G. Then the tangent groupoid $\tilde G$ defines a continuous field of C^*-algebras $t\in{\bf R}\mapsto A_t$ , with A_t=C^*(G) for $t\not=0$ and $A_0=C^*(\cal G)$ which gives the bracket of the Poisson algebra $C^\infty({\cal G}^*)$ as a limit of commutators.
For an arbitrary Lie algebroid $\cal G$ , one can only construct a local Lie groupoid G which has $\cal G$ as its Lie algebroid. The construction of the tangent groupoid has been extended to local Lie groupoids in the context of pseudo-differential operators by Nistor, Weinstein and Xu. The missing ingredient in order to extend the above result to an arbitrary Lie algebroid is the construction of the C^*-algebra of a local Lie groupoid.
Question: Is it possible to extend the usual construction of the convolution C^*-algebra C^*(G) to a local Lie groupoid G ?

Contributor: Arlan Ramsay
June 1999, Boulder Preamble. Consider the transformation groupoid $G=H\times X$ of a smooth action of a Lie group H on a manifold X. Then for each $x\in X$ , the stabilizer G(x) acts on the tangent space T_xX. This also works if the groupoid G has a local transformation group structure (union of transformation groupoids $H_i\times X_i$ with compatibility).
Question: Let G be a smooth groupoid. Can we get G(x) to act on T_xG⁽⁰⁾ without assuming the existence of a suitable choice of ``horizontal'' spaces in TG over G⁽⁰⁾?

Contributor: Jean Renault
June 1999, Boulder Preamble. Let us say that an étale groupoid G is singly generated if there exists an open s-section $T\subset G$ such that $G=\cup T^n, n\in {\bf Z}$ . For example, the semi-direct product groupoids G(X,T) associated to a local homeomorphism $T:U\subset X\rightarrow V\subset X$ are singly generated. In particular, Cuntz-Krieger groupoids and AF groupoids are singly generated. Let us say that a singly generated groupoid G is uniquely singly generated if its only generators are T (or T^-1 if T is a bisection). M. Boyle has shown that, whenever T is an essentially transitive homeomorphism of a compact space X, G(X,T) is uniquely singly generated. This is also true when T is a homeomorphism of a connected space X. On the other hand AF groupoids are not uniquely singly generated.
Question: Is there a general criterion for unique single generation?

Contributor: Dmitriy Rumynin
June 1999, Boulder Question: Let G and H be Lie groupoids over a smooth manifold X with respective Lie algebroids $\cal G$ and $\cal H$ . Given a Lie algebroid morphism $\begin{displaymath}\phi:{\cal G}\rightarrow {\cal H}\end{displaymath}$
does there exist a groupoid map
$\begin{displaymath}\Phi:\tilde G\rightarrow H\end{displaymath}$
for a covering groupoid $\tilde G\rightarrow G$ having the same Lie algebroid $\cal G$ such that $d\Phi=\phi$ ?
Comment. It may be easy or known. If it is not, then I might be able to apply my new technique to settle this. In fact, I had to integrate a Lie algebroid map to a groupoid map in my work. I did it in the context of algebraic geometry in characteristic p for certain groupoids concentrated on the first Frobenius kernel of the diagonal. Not being an analyst, I nevertheless believe that my technique would be applicable for Lie groupoids as well.
Answer proposed by Ieke Moerdijk at March 1, 2001.

Contributor: Masamichi Takesaki
June 1999, Boulder Question: Find a counterpart of the characteristic square of a factor in the context of measured ergodic groupoids or Poisson manifolds.

About this document ...

Back to Groupoid Home Page

Contributor: André Haefliger June 1999, Boulder	Question: Let $\cal F$ be a foliation on a manifold M such that the universal covering of each leaf is contractible and T be a transveral. Let G be the fundamental groupoid of $\cal F$ associated with T. Prove that for every G-sheaf $\cal A$ on T, the map $\begin{displaymath}H^(G,{\cal A})\rightarrow H^(M,{\cal A})\end{displaymath}$ is an isomorphism. Answer proposed by Ieke Moerdijk at January 22, 2001.
Contributor: Steve Hurder June 1999, Boulder	Question: Let $\cal F$ be an amenable C¹ foliation on a compact manifold M. a) If L is a compact leaf, must its holonomy group be amenable? b) Same question for a dense leaf L.
Contributor: Jerry Kaminker June 1999, Boulder	Preamble. Let G be a Lie groupoid and $\cal G$ its Lie algebroid. Then we have the pseudo-differential operators extension $\begin{displaymath}0\rightarrow C^(G)\rightarrow \Psi^0(G)\rightarrow C_0(S{\cal G}^)\rightarrow 0 \leqno{(1)}\end{displaymath}$ as an element of $KK^1(C_0(S{\cal G}^), C^(G))$ . In the case when $G=M\times M$ , the extension $\begin{displaymath}0\rightarrow {\cal K}\rightarrow \Psi^0(G)\rightarrow C_0(S^M)\rightarrow 0\end{displaymath}$ is determined by the symplectic structure of S^M via the Index Theorem for Families. Question: What does the Poisson algebra structure of $C^\infty({\cal G}^*)$ determine about (1) ? Does it determine the cohomology class needed in a topological index formula?
Contributor: Alex Kumjian June 1999, Boulder	Question: Find a groupoid cohomology (with coefficients) valid for locally compact groupoids such that the following conditions are satisfied: 1) Theory agrees with Grothendieck's equivariant sheaf cohomology if the groupoid is étale. 2) Theory agrees with Moore's cohomology if the groupoid is a locally compact group. 3) An equivalence of groupoids induces an isomorphism of cohomology. 4) The second cohomology with the circle as coefficients may be identified with the Brauer group of the groupoid.
Contributor: Alex Kumjian June 1999, Boulder	Question: Let $\phi:\Gamma\rightarrow\Lambda$ be a continuous open morphism (surjective) between groupoids. Under what conditions does one get a Fell bundle E over $\Lambda$ such that $E_\lambda$ is the closure of $C_c(\phi^{-1}(\lambda))$ (under an appropriate norm) and $\begin{displaymath}C^(\Gamma)\simeq C^(\Lambda,E) ?\end{displaymath}$
Contributor: Paul Muhly June 1999, Boulder	Question: Let G be a principal, étale groupoid. When does there exist a ``faithful'' 1-cocycle $c:G\rightarrow {\bf R}$ , i.e. a continuous homomorphism $c:G\rightarrow {\bf R}$ such that c^-1(0)=G⁽⁰⁾ ? A necessary condition is amenability, but is it sufficient?
Contributor: Chris Phillips June 1999, Boulder	Question: Let G be an amenable étale essentially principal groupoid. When is C_r^(G) a stably finite C^-algebra?
Contributor: Chris Phillips June 1999, Boulder	Question: Let G be an étale groupoid and let X be a proper G-space. Can K⁰_G(X) be described using finite dimensional vector bundles only?
Contributor: Birant Ramazan June 1999, Boulder	Preamble. Suppose that $\cal G$ is the Lie algebroid of a Lie groupoid G. Then the tangent groupoid $\tilde G$ defines a continuous field of C^-algebras $t\in{\bf R}\mapsto A_t$ , with A_t=C^(G) for $t\not=0$ and $A_0=C^(\cal G)$ which gives the bracket of the Poisson algebra $C^\infty({\cal G}^)$ as a limit of commutators. For an arbitrary Lie algebroid $\cal G$ , one can only construct a local Lie groupoid G which has $\cal G$ as its Lie algebroid. The construction of the tangent groupoid has been extended to local Lie groupoids in the context of pseudo-differential operators by Nistor, Weinstein and Xu. The missing ingredient in order to extend the above result to an arbitrary Lie algebroid is the construction of the C^-algebra of a local Lie groupoid. Question:* Is it possible to extend the usual construction of the convolution C^-algebra C^(G) to a local Lie groupoid G ?
Contributor: Arlan Ramsay June 1999, Boulder	Preamble. Consider the transformation groupoid $G=H\times X$ of a smooth action of a Lie group H on a manifold X. Then for each $x\in X$ , the stabilizer G(x) acts on the tangent space T_xX. This also works if the groupoid G has a local transformation group structure (union of transformation groupoids $H_i\times X_i$ with compatibility). Question: Let G be a smooth groupoid. Can we get G(x) to act on T_xG⁽⁰⁾ without assuming the existence of a suitable choice of ``horizontal'' spaces in TG over G⁽⁰⁾?
Contributor: Jean Renault June 1999, Boulder	Preamble. Let us say that an étale groupoid G is singly generated if there exists an open s-section $T\subset G$ such that $G=\cup T^n, n\in {\bf Z}$ . For example, the semi-direct product groupoids G(X,T) associated to a local homeomorphism $T:U\subset X\rightarrow V\subset X$ are singly generated. In particular, Cuntz-Krieger groupoids and AF groupoids are singly generated. Let us say that a singly generated groupoid G is uniquely singly generated if its only generators are T (or T^-1 if T is a bisection). M. Boyle has shown that, whenever T is an essentially transitive homeomorphism of a compact space X, G(X,T) is uniquely singly generated. This is also true when T is a homeomorphism of a connected space X. On the other hand AF groupoids are not uniquely singly generated. Question: Is there a general criterion for unique single generation?
Contributor: Dmitriy Rumynin June 1999, Boulder	Question: Let G and H be Lie groupoids over a smooth manifold X with respective Lie algebroids $\cal G$ and $\cal H$ . Given a Lie algebroid morphism $\begin{displaymath}\phi:{\cal G}\rightarrow {\cal H}\end{displaymath}$ does there exist a groupoid map $\begin{displaymath}\Phi:\tilde G\rightarrow H\end{displaymath}$ for a covering groupoid $\tilde G\rightarrow G$ having the same Lie algebroid $\cal G$ such that $d\Phi=\phi$ ? Comment. It may be easy or known. If it is not, then I might be able to apply my new technique to settle this. In fact, I had to integrate a Lie algebroid map to a groupoid map in my work. I did it in the context of algebraic geometry in characteristic p for certain groupoids concentrated on the first Frobenius kernel of the diagonal. Not being an analyst, I nevertheless believe that my technique would be applicable for Lie groupoids as well. Answer proposed by Ieke Moerdijk at March 1, 2001.
Contributor: Masamichi Takesaki June 1999, Boulder	Question: Find a counterpart of the characteristic square of a factor in the context of measured ergodic groupoids or Poisson manifolds.