Some stuff on random groups
Feb. 17th, 2019 05:41 pmSo I went to the colloquium at the math department the other week and it was Melanie Wood talking about random groups. (I've been on a more diurnal schedule since MIT Mystery Hunt so it is easier for me to make it down to the math department for colloquia and tea and such. Although what with the weather recently I had largely been staying inside and not going downtown or anything, but now it is less cold, so yay.)
So, funny thing -- when I was living in Madison I tried to go by the math department there every now and then but I rarely found anyone there, except one time I went there just after what is apparently some weekly talk they had and encountered John there. And the talk was Melanie Wood talking about random groups! Except I had just missed it, so John had to summarize it for me (also that was probably not quite the same talk).
So we start with the Cohen-Lenstra heuristics -- which not being a real number theorist I was not previously familiar with -- where you're looking at class groups of real quadratic fields and it seems like (i.e. it is conjectured that) each finite abelian group A appears with frequency proportional to 1/|A||Aut A|. Well, except not quite, because actually the 2-part has something different going on, but if you take just the odd part, each odd A occurs as the odd part with frequency proportional to 1/|A||Aut A|. Or if you just take the p-part, each finite abelian p-group A occurs as the p-part with frequency proportional 1/|A||Aut A| (well, for odd p, anyway). (Note how the automorphism group of a finite abelian group is the direct product of the automorphism groups of its p-parts.)
Anyway so Wood's big thing is how actually, this isn't because of something *special* about class groups (aside from the 2-part), but rather it's because they're essentially random. Because, see, if you make a random abelian group by taking n generators and then modding out by n+1 random relations -- you can't actually pick the relations uniformly but you can fake it in various ways -- and then you let n go to infinity, you get this distribution, where each A appears with probability proportional to 1/|A||Aut A|. (You could also get something infinite, but with probability 1 you get something finite.)
And she even has a stronger version of this theorem -- she calls it a central limit theorem for finite abelian groups -- where in fact you don't have to pick the random relations in a uniform way, but in fact pretty much any way, so long as each coordinate is independent, and there's no coordinate which is completely concentrated in a single residue class mod some prime.
But the thing is, it's not really that class groups are random finite abelian groups (ignoring 2 anyway), but that they're random finite abelian groups with one more relator than generator. Because instead of using n+1 relations, you could use n+u relations. And then, for u>0, it's proportional to 1/|A|u|Aut A| instead.
She actually didn't talk about that during the talk itself but me and someone else asked her about it later. You can do u<0 as well, then of course you won't get something finite, rather (I think, if I'm understanding this right) with probability 1 you get something of rank -u (because duh) but its torsion part is just like above, it's proportional to 1/|A|-u|Aut A|.
Now where it gets weird is u=0. I was asking Melanie about this at the dinner afterward and, so, uh, limits of probability distributions are weird. Because that's the thing, right? It's a limit as n goes to infinity. Here's an example she gave to illustrate this: Let's consider the probability distribution νn on [0,1] given by, each rational number in [0,1) of denominator dividing n gets probability 1/n. What's the limit of these distributions as n→∞? It's the Lebesgue measure! I mean, I guess that makes perfect sense if you look at it from the point of view of, what is the actual measure of actual sets, but it still seems pretty crazy in some sense because, like, you're starting with just rational numbers, but you get this distribution that's uniform on all of [0,1]. And the rationals have measure 0 in it -- all the measure has sort of "escaped" onto these new irrationals that have popped up. Probability is weird like that.
Anyway so something similar happens in the u=0 case -- with probability 1 you get an infinite group, but they're profinite groups (even though those aren't, like, actual groups you could get before you took a limit). If you look at just the p-part, though, then this is a finite abelian p-group A with probability proportional to 1/|Aut A|.
This really confused me at first because I was thinking in terms of, y'know, groups you could actually get, finite groups, and I was like, hold on, the sum of 1/|Aut A| doesn't converge (consider if A is Cp), there's no constant of proportionality you can use there to make it a probability distribution! But like... it doesn't work like that. You get these infinite profinite groups. But the p-parts work fine. So regardless of the value of u, a given p-part A appears with probability proportional to 1/|A|u|Aut A|. (Notice how if you restrict A to be p-groups for a fixed p, this now does converge even when u=0.)
Although really the main part of her talk was not all this abelian stuff, but rather the more difficult nonabelian case. Which from what John told me was what she was talking about that time I just missed her, too. The nonabelian case, profinite groups get involved even for u>0, and it's much more complicated; the probability of getting a finite group is nonzero, but so is the probability of getting an infinite group. Also you can't actually get all finite groups because not all of those can be expressed with just one more relation than generator. So I guess for any fixed u you'll miss stuff. And when you look at the probabilities you get these factors corresponding not just to primes but also to nonabelian finite simple groups. And of course this comes from a unified abstract thing where these are both just, y'know, finite simple groups, but apparently when you make this more concrete the factors for the nonabelian simple groups end up looking different from the factors for the abelian ones. IDK. (John was telling me about the thing with the finite simple groups that other time, FWIW.) Also Wood did explicitly note that this theorem did not depend on the classification of finite simple groups, although of course doing any concrete computations with it does.
Btw even though the probability of getting a finite group is neither 0 nor 1 Wood didn't give any approximation of it; apparently she hadn't really attempted to compute it. She did give the approximate probability of getting the trivial group (about 43%). She said to get the probability of getting a finite group you'd just have to sum the probabilities for individual finite groups... still, I would have expected that as the groups get larger the probabilities would drop off pretty fast so you wouldn't have to sum that many for a good approximation. IDK. Maybe they just don't drop off that fast. That would make sense I guess.
Note that Wood doesn't have to my knowledge any "central limit theorem" for the nonabelian case; I think this is all assuming uniformity (or fake uniformity). Also the nonabelian case doesn't match this particular noncommutative analogue of the Cohen-Lenstra heuristics that she was considering (though I didn't quite get where that was coming from), but she thinks it can be modified to match.
Anyway what I'm wondering about now is (because I didn't think to ask her about this at the time) -- going back to the abelian case, what if we did this over other rings? Like let's say a PID or maybe more generally a Dedekind domain, to keep things nice. So instead of finite abelian groups, you have torsion R-modules. I guess the thing is that in general since these wouldn't be finite, 1/|A|u|Aut A| doesn't make a lot of sense anymore? I guess you could restrict specifically to the case of global fields so that they would be finite, maybe it'd work then, IDK.
Maybe I should write to her and ask about this, or maybe I should just try to look it up...
-Harry
So, funny thing -- when I was living in Madison I tried to go by the math department there every now and then but I rarely found anyone there, except one time I went there just after what is apparently some weekly talk they had and encountered John there. And the talk was Melanie Wood talking about random groups! Except I had just missed it, so John had to summarize it for me (also that was probably not quite the same talk).
So we start with the Cohen-Lenstra heuristics -- which not being a real number theorist I was not previously familiar with -- where you're looking at class groups of real quadratic fields and it seems like (i.e. it is conjectured that) each finite abelian group A appears with frequency proportional to 1/|A||Aut A|. Well, except not quite, because actually the 2-part has something different going on, but if you take just the odd part, each odd A occurs as the odd part with frequency proportional to 1/|A||Aut A|. Or if you just take the p-part, each finite abelian p-group A occurs as the p-part with frequency proportional 1/|A||Aut A| (well, for odd p, anyway). (Note how the automorphism group of a finite abelian group is the direct product of the automorphism groups of its p-parts.)
Anyway so Wood's big thing is how actually, this isn't because of something *special* about class groups (aside from the 2-part), but rather it's because they're essentially random. Because, see, if you make a random abelian group by taking n generators and then modding out by n+1 random relations -- you can't actually pick the relations uniformly but you can fake it in various ways -- and then you let n go to infinity, you get this distribution, where each A appears with probability proportional to 1/|A||Aut A|. (You could also get something infinite, but with probability 1 you get something finite.)
And she even has a stronger version of this theorem -- she calls it a central limit theorem for finite abelian groups -- where in fact you don't have to pick the random relations in a uniform way, but in fact pretty much any way, so long as each coordinate is independent, and there's no coordinate which is completely concentrated in a single residue class mod some prime.
But the thing is, it's not really that class groups are random finite abelian groups (ignoring 2 anyway), but that they're random finite abelian groups with one more relator than generator. Because instead of using n+1 relations, you could use n+u relations. And then, for u>0, it's proportional to 1/|A|u|Aut A| instead.
She actually didn't talk about that during the talk itself but me and someone else asked her about it later. You can do u<0 as well, then of course you won't get something finite, rather (I think, if I'm understanding this right) with probability 1 you get something of rank -u (because duh) but its torsion part is just like above, it's proportional to 1/|A|-u|Aut A|.
Now where it gets weird is u=0. I was asking Melanie about this at the dinner afterward and, so, uh, limits of probability distributions are weird. Because that's the thing, right? It's a limit as n goes to infinity. Here's an example she gave to illustrate this: Let's consider the probability distribution νn on [0,1] given by, each rational number in [0,1) of denominator dividing n gets probability 1/n. What's the limit of these distributions as n→∞? It's the Lebesgue measure! I mean, I guess that makes perfect sense if you look at it from the point of view of, what is the actual measure of actual sets, but it still seems pretty crazy in some sense because, like, you're starting with just rational numbers, but you get this distribution that's uniform on all of [0,1]. And the rationals have measure 0 in it -- all the measure has sort of "escaped" onto these new irrationals that have popped up. Probability is weird like that.
Anyway so something similar happens in the u=0 case -- with probability 1 you get an infinite group, but they're profinite groups (even though those aren't, like, actual groups you could get before you took a limit). If you look at just the p-part, though, then this is a finite abelian p-group A with probability proportional to 1/|Aut A|.
This really confused me at first because I was thinking in terms of, y'know, groups you could actually get, finite groups, and I was like, hold on, the sum of 1/|Aut A| doesn't converge (consider if A is Cp), there's no constant of proportionality you can use there to make it a probability distribution! But like... it doesn't work like that. You get these infinite profinite groups. But the p-parts work fine. So regardless of the value of u, a given p-part A appears with probability proportional to 1/|A|u|Aut A|. (Notice how if you restrict A to be p-groups for a fixed p, this now does converge even when u=0.)
Although really the main part of her talk was not all this abelian stuff, but rather the more difficult nonabelian case. Which from what John told me was what she was talking about that time I just missed her, too. The nonabelian case, profinite groups get involved even for u>0, and it's much more complicated; the probability of getting a finite group is nonzero, but so is the probability of getting an infinite group. Also you can't actually get all finite groups because not all of those can be expressed with just one more relation than generator. So I guess for any fixed u you'll miss stuff. And when you look at the probabilities you get these factors corresponding not just to primes but also to nonabelian finite simple groups. And of course this comes from a unified abstract thing where these are both just, y'know, finite simple groups, but apparently when you make this more concrete the factors for the nonabelian simple groups end up looking different from the factors for the abelian ones. IDK. (John was telling me about the thing with the finite simple groups that other time, FWIW.) Also Wood did explicitly note that this theorem did not depend on the classification of finite simple groups, although of course doing any concrete computations with it does.
Btw even though the probability of getting a finite group is neither 0 nor 1 Wood didn't give any approximation of it; apparently she hadn't really attempted to compute it. She did give the approximate probability of getting the trivial group (about 43%). She said to get the probability of getting a finite group you'd just have to sum the probabilities for individual finite groups... still, I would have expected that as the groups get larger the probabilities would drop off pretty fast so you wouldn't have to sum that many for a good approximation. IDK. Maybe they just don't drop off that fast. That would make sense I guess.
Note that Wood doesn't have to my knowledge any "central limit theorem" for the nonabelian case; I think this is all assuming uniformity (or fake uniformity). Also the nonabelian case doesn't match this particular noncommutative analogue of the Cohen-Lenstra heuristics that she was considering (though I didn't quite get where that was coming from), but she thinks it can be modified to match.
Anyway what I'm wondering about now is (because I didn't think to ask her about this at the time) -- going back to the abelian case, what if we did this over other rings? Like let's say a PID or maybe more generally a Dedekind domain, to keep things nice. So instead of finite abelian groups, you have torsion R-modules. I guess the thing is that in general since these wouldn't be finite, 1/|A|u|Aut A| doesn't make a lot of sense anymore? I guess you could restrict specifically to the case of global fields so that they would be finite, maybe it'd work then, IDK.
Maybe I should write to her and ask about this, or maybe I should just try to look it up...
-Harry
