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Can somebody provide an example of a locally compact groupoid $G$ with a Haar system such that the range map restricted to isotropy groupoid of $G$ is open?

I could not find any specific example for that. I just want to clarify whether the action groupoid $G= H\ltimes X$ where $H$ is a locally compact group and $X$ a locally compact space is an example for this.

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  • $\begingroup$ What is a Haar system? $\endgroup$
    – LSpice
    Commented Jun 7 at 13:43
  • $\begingroup$ Presumably you want to eliminate silly cases like when the isotropy subgroupoid is open eg for discrete groupoids or groupoids with only isotropy $\endgroup$
    – Benjamin Steinberg
    Commented Jun 7 at 23:29
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    $\begingroup$ @LSpice, Haar systems are the groupoid analogues of Haar measure. They are not automatic and when they exist the range map is open $\endgroup$
    – Benjamin Steinberg
    Commented Jun 7 at 23:30
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    $\begingroup$ @LSpice a suitably invariant family of measures on the source fibres $s^{-1}(x)$ (say) of a suitable topological groupoid (for all $x$ in the space of objects), or more generally something like a groupoid internal to measurable spaces. $\endgroup$
    – David Roberts ♦
    Commented Jun 13 at 1:05

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In the case of the groupoid G=H⋉X, if the action is free (H⋉X is a principal groupoid), then the range map restricted to isotropy groupoid is open. In general it is not true.

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  • $\begingroup$ Welcome to MO, Prof Buneci! $\endgroup$
    – David Roberts ♦
    Commented Jun 12 at 22:31

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