 Okay, great. Thank you all for coming out on a Saturday and thank you to the organizers for giving me the opportunity to speak. So, today I want to talk about how we can use and how we can understand torsion in big mapping class groups. So, all of this work is going to be joint with Danny Caligari, Lujo Chun, and Rayleigh Lyman, everything that I say today. So, the large overarching question that I want to talk about is understanding the structure of subgroups within big mapping class groups. So, what kinds of groups can we find as a subgroup of big mapping class groups? So, before we dive into this interesting question and story, I want to just lay the groundwork for what is a mapping class group and why are some of them big and some of them not. So, to first start, let's talk about the mapping class group. This is associated to a surface. So, it's a two-manifold and we're going to look at the self-homomorphisms of that surface that preserve orientation and then we're just going to kind of fudge things and say, oh, we only want them up to homotopy. So, we're just thinking about really nice things. Unfortunately, this forms a group and it's very classically studied. I think the first lecture that Max Dane gave on mapping class groups was, I think, 99 years ago. So, we're almost at 100 years. It's a deep, rich, very mysterious group that shows up all over math in dynamics, algebraic geometry, two, three, four-manifold theory, great stuff. But typically, we only focus on surfaces that are algebraic or that have a finitely generated fundamental group. So, I want to talk about surfaces that don't satisfy this, surfaces with a fundamental group that's not finitely generated. These surfaces either have infinite genus or they'll have infinitely many punctures on their surface. So, just to kind of give you a taste for what the zoo of infinite type surfaces can look like, I'll give you all some examples. So, the first maybe kind of naive example is just an infinite connect sum of to or I often to infinity. So, this has infinite genus and one end, one way of escaping off to infinity. This is called the Lochus monster. This surface here is called the flute surface, I guess because it sort of looks like a flute. It's homeomorphic to, if you were to take the complex plane and then lift out the integers, you would get something homeomorphic to the flute surface. So, this is genus zero, infinitely many punctures. And then one of my favorites, this is the blooming canter tree. So, this is an infinite genus surface with a canter set worth of ways of flowing off to infinity. So, we know a lot about classical mapping class groups and we'd like to know, you know, these mapping class groups of the infinite type surfaces would be called big. And so, certainly, there should be some kind of distinction, otherwise we wouldn't just say big, we would say mapping class groups. So, let's talk about some of the distinctions. For this slide, I'm going to be throwing out maybe some algebraic jargon if you don't know the particulars. That's not particularly important here. Just to get a sense of how these things can be different. So, mapping class groups of finite type surfaces are typically, they're all finitely generated. But once you move into the infinite type land, these are actually uncountably infinite groups. They're these huge groups. And so, immediately, because of their cardinality, they're kind of barred from the land of finite generation. Mapping class groups are also classically, residually finite. But in 2017, Patel of Llamas showed that no big mapping class groups can be residually finite. The key reason here is because there's actually a braid group on infinite strands sitting as a subgroup of big mapping class groups. And those aren't residually finite. And lastly, mapping class groups usually satisfy something called the tits alternative. And there's been a lot of activity on this, showing that big mapping class groups do not satisfy the tits alternative. They don't enjoy this algebraic property. Lanier and Loving last year showed that they don't satisfy the strong tits alternative. In a paper that was just released two weeks ago, actually, Algar Patel and Llamas showed that if you have infinite genus, then your mapping class group doesn't satisfy this alternative. And in a forthcoming paper of Alcock, Alcock shows that, in fact, this is just true for any infinite type surface. Their mapping class group doesn't satisfy this alternative. So this definitely seems like a breakaway from traditional mapping class groups. And then in the Algar Patel of Llamas paper, they show this. They have this theorem. And I should say before I say anything about it, this is actually a weaker version that I have up here. This is a weaker version of the theorem that they actually prove in their paper, which is kind of a remarkable statement because this is a wild theorem to me. The theorem is that there's these large infinite families of big mapping class groups that either have every single finite group as a subgroup, or every single countable group as a subgroup. This is wild. I come from a background of geometric group theory, where many of the groups that I care about are countable. And somehow every single one of them is a subgroup of big mapping class groups. If you're not familiar with mapping class group theory, there's this really beautiful tension between the rigidity of these finite type surfaces and the structure of mapping class groups. And so far, we're seeing something that seems kind of bad for big mapping class groups that maybe they don't have quite enough structure to satisfy that interesting mysterious edge that mapping class groups of finite type surfaces have. But this is only true if you are kind of solely thinking about groups from this countable, discrete sort of perspective like we like to do often in geometric theory. However, big mapping class groups come with a very interesting, rich, natural topology associated to them, and they're also uncountable. So they don't fit into this countable, discrete landscape. So I think if we can kind of push on the topology of the group, thinking about it as a topological group, then I think maybe we might find some interesting mathematics that can happen here. So let's do this. Let's kind of talk about the topology. So there were two sources of motivation that me and my collaborators had when we got to thinking about the actual topology structure on big mapping class groups. The first one was the Nielsen realization problem. This is actually a theorem for finite type surfaces. But the question is, if I have a finite subgroup of my mapping class group, can I lift it into the isometry group of some complete hyperbolic metric on my surface? There's a very celebrated theorem of Kirchhoff that indeed this is true for finite type surfaces. And we really wanted this to be true for infinite type surfaces. What this means kind of maybe intuitively is that these the finite order elements kind of look like these rigid rotations. And we didn't want any like strange twisty torsion elements for infinite type surfaces. So we wanted to prove this. And because of Aga, Patel, and Vlamas' result that we have these large infinite families of mapping, big mapping class groups that contain every single countable group as a subgroup, we didn't want this just to be a result of like Cayley's theorem. If I have a countable group, then it embeds into bijections on an accountable set. We wanted that result to be really something richer and more deep about the web of subgroups. So we wanted to kind of bar the symmetric group of the natural numbers from being a subgroup. So we wanted to prove this thing about isometries and we wanted to bar this particular group. So the first thing that we did is we just proved Nielsen realization. So the same thing that is true for finite type surfaces is indeed true for infinite type surfaces. This took a little bit of work of stitching together Kirchhoff's proof for little finite type pieces of our surface. But this is kind of nice. It's saying that the torsion elements that you might want are exactly what you expect. And then we were thinking about the symmetric group. And this is really where the topology of the group started to enter our minds. And I want to just take a moment to thank Katherine Mann. She suggested that we should really think about the topology of the symmetric group and how it may or may not be compatible with the topology of a big mapping class group. So here's an interesting fact about the symmetric group is that there are no torsion free neighborhoods of the identity. There are sequences like I've given here of finite order elements that limit to the identity. In this case, it's an order two element for each term in the sequence and you're just stripping away transpositions. So for every natural number, there's some point in the sequence where you're just eventually fixed forever. So this limits the identity. So we were thinking about these two facts. We had Nielsen realization. So these finite order elements are isometries. And we were wondering if this approximating the identity concept was also going to be true in the mapping class groups. And then we stumbled on just one kind of simple fact. If you take a pair of pants in the surface, so this is what I've drawn in green on the right hand side here. It's a sphere with three boundary components. If I have an isometry and it fixes each of the boundary curves individually, then I actually have to be trivial. There's no non-trivial isometry that can fix a pair of pants. At least fix the cuffs individually. So if you know something about the topology of big mapping class groups, it's actually that the fact that the symmetric group isn't a subgroup kind of pops out immediately. So let me say a little bit about what the topological structure is. If you're not familiar with mapping class groups, this is an important thing to know. Mapping class group theorists love acting on curves. This is one of our favorite things to do. We put curves into complexes, act on that, and we love studying that kind of an action. So in true mapping class group form, we're going to do the same thing. We're going to have our big mapping class group act on these homotopy classes of curves. And what we're going to do is take for each curve, you have a subgroup of the big mapping class group that just stabilizes that curve. So we're going to gather up all of those stabilizer subgroups and just snap our fingers, declare them to be open sets in our group, and then look at the courses topology that that collection of sets generate. So that's the topology. So anytime you see your stabilizing curves, you should be thinking, okay, we actually have an open set here. So if we come back to this idea of the pair of pants, once we fix each of these three curves individually, that's just the intersection of three stabilizer subgroups. And that actually gives us a neighborhood of the identity. But we said that there can't be any isometry. So this is an isometry free neighborhood of the identity. But we have Nielsen realization, which says that torsion and is our isometries. So here, not only is this a isometry free zone, it is also a torsion free zone. But for the symmetric group, this isn't this isn't true, we can find that you can never find a torsion free neighborhood of the identity. So automatically, we get that the that this infinite symmetric group isn't a subgroup for the big math and class group. And really, this wasn't specific to just the symmetric group. It could be any group that has no torsion free neighborhood of the identity. And in fact, we can push this just a little bit further. So we came up with two additional topological obstructions that I'm personally really excited about. And they all stem from this fact that there are torsion free neighborhoods of the identity in it in any big mapping class group. The first is that if you have a compact subgroup, you have to be a finite group. And that locally compact subgroups are discreet. And I just want to mention, thank you to one that's been out for for suggesting that these ought to be true in our context. What I find particularly interesting about this is that now we have a pretty large dichotomy between what our subgroups look like within big mapping class groups, if we want to think topologically. People who study topological groups know that locally compact groups are really nice. This is any any league group, right, which is, you know, a positive dimension is locally compact. And what this theorem is telling us is that there's actually no league groups sitting inside of our big mapping class groups. And indeed, if I give you a subgroup, I know that either it has to be discreet, or it has to be some large, non locally compact group. So I think finally, we're starting to kind of get down to some of the interesting pieces of that this interesting web of subgroups big, big, big mapping class groups have. So I'd like to end there. Thank you all for listening.