By "irresponsible speculation", I mean "speculation without having done my homework first", i.e., without having even tried to look at the existing literature beyond this paper or actually really tried anything at all.
So, Juan Arias de Reyna recently pointed out to me the following paper: Commuting Probabilities of Finite Groups, by Sean Eberhard.
EDIT DEC 1: The paper has been updated; the following refers to the original version of the paper. See the comments for discussion of the updated version.
So: If you have a finite group, you can consider the probability that a randomly chosen ordered pair of elements from it commutes. You could then consider the set of all probabilities obtained this way; this is some set of rational numbers between 0 and 1.
In this paper, Eberhard shows that this set is in fact reverse well-ordered! Its order type is of the form ωα where 2≤α≤ω². (Though Juan Arias de Reyna points out that it's not too hard to find examples that show that in fact α≥ω, so the order type is at least ωω.) He also shows that all the limit points of this set are rational. I think it should be pretty obvious why I'm interested in this! (Even though I have no plans to actually study this, my hands are full as it is.)
Now for the irresponsible speculation:
1. Instead of just doing finite groups, one could generalize to compact groups; one can consider probabilities there as well. How would this affect the set? It would be really nice if this just gave you the closure of the probability set for finite groups! Though Arias de Reyna tells me it's conjectured that the finite group commuting probability set is closed in (0,1], so that would be saying that using general compact groups only gives you 0 in addition. (I mean, it certainly does give you 0 in addition!)
2. I'm reminded of some work of John Wiltshire-Gordon and Gene Kopp. They considered probabilities of randomly chosen elements from a compact group satisfying some general word; the case of commuting probabilities is the case where the word w is aba-1b-1. I wonder if the same phenomenon might be seen for other words.
Probably it would be best to first look at words in one variable. Obviously using w=a generates an ω if we stick to finite groups and an ω+1 if we allow general compact groups -- not ωω, but still well-ordered. As for a² or a³... well, I don't know, and I don't really intend to work on it as I said above, but it seems vaguely plausible and it's an interesting question!
So yeah that's basically all I have to say on the matter.
-Harry
So, Juan Arias de Reyna recently pointed out to me the following paper: Commuting Probabilities of Finite Groups, by Sean Eberhard.
EDIT DEC 1: The paper has been updated; the following refers to the original version of the paper. See the comments for discussion of the updated version.
So: If you have a finite group, you can consider the probability that a randomly chosen ordered pair of elements from it commutes. You could then consider the set of all probabilities obtained this way; this is some set of rational numbers between 0 and 1.
In this paper, Eberhard shows that this set is in fact reverse well-ordered! Its order type is of the form ωα where 2≤α≤ω². (Though Juan Arias de Reyna points out that it's not too hard to find examples that show that in fact α≥ω, so the order type is at least ωω.) He also shows that all the limit points of this set are rational. I think it should be pretty obvious why I'm interested in this! (Even though I have no plans to actually study this, my hands are full as it is.)
Now for the irresponsible speculation:
1. Instead of just doing finite groups, one could generalize to compact groups; one can consider probabilities there as well. How would this affect the set? It would be really nice if this just gave you the closure of the probability set for finite groups! Though Arias de Reyna tells me it's conjectured that the finite group commuting probability set is closed in (0,1], so that would be saying that using general compact groups only gives you 0 in addition. (I mean, it certainly does give you 0 in addition!)
2. I'm reminded of some work of John Wiltshire-Gordon and Gene Kopp. They considered probabilities of randomly chosen elements from a compact group satisfying some general word; the case of commuting probabilities is the case where the word w is aba-1b-1. I wonder if the same phenomenon might be seen for other words.
Probably it would be best to first look at words in one variable. Obviously using w=a generates an ω if we stick to finite groups and an ω+1 if we allow general compact groups -- not ωω, but still well-ordered. As for a² or a³... well, I don't know, and I don't really intend to work on it as I said above, but it seems vaguely plausible and it's an interesting question!
So yeah that's basically all I have to say on the matter.
-Harry
