Skip to main content
11 events
when toggle format what by license comment
Feb 22, 2021 at 16:53 history became hot network question
Feb 22, 2021 at 13:13 answer added Benjamin Steinberg timeline score: 5
Feb 22, 2021 at 12:57 comment added Ville Salo Glad to hear it's not completely trivial and that generators have come up, I am looking forward to a prospective semianswer.
Feb 22, 2021 at 12:50 comment added Benjamin Steinberg As far as generators go things are complicated. There were some earlier papers that mention the issues but I forget which ones. I think it comes up in Coulbois's papers on profinite topology circa 2000 and maybe in Lawson Kellendonk but I can't quite remember. Or maybe in a later survey by Mark Lawson you might ask him. The I issue is that a partial action is not determined by what it does on generators because it isn't given by a homomorphism into the symmetric inverse monoid but instead by what is called the symmetric inverse monoid. This is getting long so I may give a semianswer
Feb 22, 2021 at 12:44 comment added Benjamin Steinberg You should look for inverse semigroup instead of pseudogroups. You'll see that there is lots of literature on partial group actions and inverse semigroups starting with the work of Exel @YCor mentioned. A good early introduction is the IJAC paper of Lawson and Kellendonk which also discusses discusses globalizing partial actions. They don't speak of your particular partial action so far as I remember.
Feb 22, 2021 at 9:32 comment added Ville Salo So you are saying, I can just say $G$ partially acts on itself by conjugation, and we take the partial subaction on $H$, and then I just say $G$ is finitely-generated. Maybe that is indeed the good setting. But I would like to say that "$G$ partially acts on $H$", and only have the conjugation partial action as a special case; the issue is, what is the big group where $G$ acts? Anyway, maybe I just need to think about this more...
Feb 22, 2021 at 9:27 comment added Ville Salo "it is natural enough to say that a group action $G \curvearrowright H$ is finitely generated when $G$ is" I mean this to be the definition. It's just a property of the group, and I'm ok with just saying that $G$ is finitely-generated, my point was just that that's already non-trivial e.g. in the groupoid approach.
Feb 22, 2021 at 9:24 comment added YCor About finite generation: for a genuine action, what would you mean by "finitely generated"?
Feb 22, 2021 at 9:23 history edited Francesco Polizzi CC BY-SA 4.0
added 10 characters in body
Feb 22, 2021 at 9:23 comment added YCor There's a good formalism of partial action of a group $G$ on a set $X$ due to Exel (each group element acts as a partial bijection of $X$, that is, a bijection between two given subsets of $X$, with natural axioms), and in particular, if $G$ acts on a set $M$ and $X$ is an arbitrary subset of $M$, then $G$ naturally partially acts on $X$ by restriction. Orbits of partial actions are well-defined (here these are just intersections of orbits on $M$ with $X$).
Feb 22, 2021 at 8:52 history asked Ville Salo CC BY-SA 4.0