 Thank you very much. So here's the plan of the talk. We are going to start with an introduction to HHSs. And so then we are going to talk about their boundaries. And there's various things you can do with the boundary of an HHS. The thing I decided to mostly focus on today is some notion of geometrical finiteness. So introduction to HHSs. So let me start. Okay. First of all, I will not give the full definition because the full definition is very long. But I will explain some of the, you know, the most interesting features of the geometry of an HHS. So let me start with a list of examples to try and get you interested in this. Because the, I mean, the examples are groups that you should care about. So examples and spaces that you should care about. The simplest examples, kind of the complexity zero examples are hyperbolic groups. And spaces in fact, but let's just say groups. And this class is closed under direct product and free products. But more generally, relative hyperbolicity. So in particular, every total relative hyperbolic group is also an HHS. More intriguingly, and that's going to be our main example, mapping class groups, our HHS. So that's, I expect most people to be interested in this. But not just mapping class groups. Some relatives of mapping class groups, very HHSs. So tecumular space. Space. With either the, there's at least two natural metrics. With either the tecumular metric, thank you. Or the Wild Peterson metric. And another class of spaces that are interesting are cut zero cube complexes. I won't say exactly which ones are HHSs, but many of them are. So for example, universal covers of compact special cube complexes. So in particular, right angle arting groups, right angle coxer groups. Universal covers of compact, non positively curved cube complexes. Compact special, non positively curved cube complexes. And let me mention the last one. Fundamental groups of non geometric three manifolds. I hope that this list of examples is extensive enough to get you interested. And so, and as I said, I will not explain the definition, but I will take what's, let's say the main example is mapping class groups. So this class of groups was inspired by machinery developed by Mazer and Minsky to study mapping class groups. And somehow the point is to extend that machinery to all the groups on the board. The initial point was to extend that machinery to all the groups on the board. So our working example is the mapping class group. And I'm going to explain to you what Mazer and Minsky did of S where S is, let's say it's closed oriented surface of genus G at least still. Okay, so the idea, the idea that one way to state what Mazer and Minsky did is that you want to study the geometry of, well, the catagraph of this group. And the way you're going to do that is to reduce the study of the geometry of the mapping class group to that of a family of hyperbolic spaces, a certain family of hyperbolic spaces. Hyperbolic spaces are kind of easy to work with, so we like them. The mapping class group is not hyperbolic, but let's try to, you know, to make it as hyperbolic as possible. Okay, so let me, what are these, what are these hyperbolic spaces going to be? They're going to be curved graphs. So let me define the curve graph of the whole surface. There will be other, other curve graphs floating around. So CS is the curve graph, and it has vertices are, in short, simple closed curves. So to say it, to say it correctly, I should say isotopic classes of essential simple closed curves. Did I forget some adjectives? And, and edges correspond to disjoint curves, or more precisely curves that can be realized disjointly. Okay, so that's a graph. It has, so as any graph has its metric, its graph metric. And the theorem of Mr. Minsky now has like much easier proofs than the original one. It's that CS is hyperbolic. Okay, so how do we compare the geometry of this graph with that of the mapping class group? We have a, we have a map. So the mapping class group acts on the curve complex, which then gives us a map, which is the orbit map. Okay, so the stabilizers are very big. So, you know, this action is not proper, but, so it doesn't encode the whole geometry of the mapping class group. All right, I should, let me say, let me say something first. So, as it turns out, it's more convenient to define, to, to fix a fancy comp, so instead of fixing a, a point in CS, we fix a fancy composition, which is a maximal simplex in CS. Right, so I need to cursor disjoint. The diameter is one. So, fix fancy composition P and pi S of some G is G of P as a subset of CS. So, this thing as diameter one. So, for our purposes, it's as good as a point. We're doing course geometry. Points and bonded sets are the same. We have a course to leave sheet mapped to a hyperbolic space. And as I said, this doesn't see the whole thing. It doesn't see the whole picture. So, remark. Let me write it informally. So, CS doesn't see what happens in proper subsurface. So, if you have the stabilizer of a curve in the mapping class group, it's, it's a pretty big subgroup, but it has bounded orbits. Okay, so, so we need not just a curve complex of the main surface, but also of proper subsurface. So, for Y a subset, a subsurface of S, one can define the curve complex of Y. And so, we are, we'll ignore, so there are some issues in low complexity, like for annual eye, one puncture towards four puncture shields. We're going to ignore that. So, we'll ignore low complexity special cases. So, the definition has to be changed in those cases, but it doesn't matter for us. Ignore low complexity cases. Okay. So, you have CY, which is defined similarly. You need to add non-peripheral. So, non-parallel to the boundary to the list of objectives, but that's a good definition. CY is still hyperbolic. And so, now, we had, we had that map from, which came from the, the orbit map from the mapping class group to CS. How do we, we also want a map to, to CY, to compare, once again, to compare the geometry to, to see, to study the geometry. What happens in that subsurface? And so, and so the map is defined as follows. So, for Q, a fancy composition, one can define the subsurface projection of Q to be informally the intersection and more formally the set of curves in Y obtained as follows. So, all the curves that resemble the intersection in some sense. So, let me draw a picture. Okay. So, let's say that this right part is the, is Y. And we're going to take some, some curve in a fancy composition. So, this is some curve in a fancy composition. And then, we are going to find kind of the simplest possible curves that are disjoint from it. So, we take a neighborhood, we take a union of this intersection and the neighborhood, sorry, we take a union of this arc and the neighborhood and the boundary and we take a neighbor. I can do that. Okay. So, it's kind of a natural way of projecting a fancy composition to a subsurface. And this also gives us a map from the mapping class group to CY. This also gives us a map, which we still denote. So, the map we still denote by Y from the mapping class group to CY. And, right. So, now we have some space that's supposed to encode what happens on any given subsurface. So, we should be able to control the whole geometry of the mapping class group using those CYs. And that is correct. And one striking way in which this is through is the distance formula, formula. Again, Mr. Minsky. And so, there are some quantifiers here. And what it says is that to measure the distance between two elements of the mapping class group, what you have to do is you have to project them to all subsurfaces. I should say all isotopic classes of subsurfaces and measure their distance there. Okay. So, this formula doesn't make sense yet, because this might be an infinite number. So, we are just going to ignore small terms. So, that symbol, let me tell you what that symbol means. So, that symbol just means that we take, so we keep A if A is at least L and 0 otherwise. Okay. Sorry, it's not equal. It's approximately equal up to multiplicative and additive error. So, like in the definition of quasi isometry. Otherwise, yeah, this depends on a generating set. Okay. And so, before I write the quantifier, let me comment on this. So, this is great. So, you can see what happens in every curve complex. Those are hyperbolic spaces. So, you have, you know, a lot of things. You know that it's hard to travel far from a JDZ. And then, you know, put, so, this formula put all this information back into information on the mapping class group. So, the quantifiers are for every threshold large enough. There exists multiplicative and additive constancy then in that symbol so that, you know, for any two elements of the mapping class group that holds. So, that's really useful. And now let me, now there's enough, I explained enough material about the mapping class group to tell you a bit about what happens for general hierarchical hyperbolic spaces. I want to keep it, there should be enough space here. Okay. So, what's the general? Yeah. It's not obvious that it's fine. It's part of the, I mean, part of the theorem is that that sum is fine. Right. I mean, there will be finitely many, finitely many terms here that are relevant and anything that's kind of complicated compared to all those surfaces we'll have a zero term. I mean, uniformly bounded term. Okay. So, the general HHS setup. In general, HHS setup, you have some Jodesic or, well, it's quasi-Jodesic space. And part of the data is a collection of hyperbolic spaces. So, collection of hyperbolic spaces with uniform constant and maps to compare X to each CY. Third course, lip sheets. It's not necessary, but you can always assume it that these are also coarsely surjective. I mean, otherwise, you just take smaller hyperbolic spaces. Okay. And so, from the setup, say, for an HHS, you have a distance formula that treats exactly the same as that distance formula there. Okay. So, in each of those spaces, each of those spaces in the list comes with a collection of hyperbolic spaces with maps and a distance formula holds. So, let me tell you a nice consequence of the distance formula that's also related to another piece of information that I want to add there. Consequence of the distance formula is that if you take two disjoint subsurfaces, then there's a natural embedding of the mapping class group of Y times the mapping class group of Z into the mapping class group of S. Okay. So, if you're given a mapping class in Y, you extend it to the identity and so on. And the point is that this natural embedding is quasi-zometric. It's quasi-zometric. So, why is that? Because the relevant distance formula terms live in, I mean, subsurfaces that are contained in Y and same for the other one. So, they kind of don't interact. So, in an HHS, you have natural product regions also and kind of to state this, so, what is disjointness and what is contained. So, these are actually part of the data that you have. So, part of the data is that there's a orthogonality relation. So, we don't call it disjointness because it would be very confusing in all other examples except for the mapping class group. So, this is disjointness. So, the reason why it's orthogonality is because it corresponds to product regions and nesting relation. And orthogonality in, I mean, very similar sentences, orthogonality corresponds to product regions. So, let me just write it formally. So, orthogonality yields product regions. Right. Yeah. So, relation, thank you. So, these are on this S, this is in excess. Yeah. Thank you. Good. On the subsurfaces. Okay. So, I think that's all I wanted to tell you about the general, for the introduction to HHS, except today it's propaganda time. HHS propaganda. So, because you might say, well, okay, I only care about the mapping class group. Why do I want to do more general things? Okay. So, one thing I'd like to tell you is that the distance formula for HHS is actually not part of the axioms, but it's a theorem. So, from the first definition, from the first version of the definition, we simplified them a lot. So, if you simplify the axioms a lot, and now we don't have to assume the distance formula. We have to assume something much bigger, which is much easier to prove. So, in particular, it's not hard. So, it's not hard to show that the mapping class group is an HHS, given that we now know how to prove easily that the curve complex is hyperbolic. So, and this gives a proof of the distance formula in particular. So, but also it's not hard to show that mapping class group or other spaces are HHS. So, and this is very relevant for the following theorem that I will not only state and not discuss much. So, this is a theorem we proved with Jason Bersack and Mark Hagen. I should mention that we use the estimate of the asymptotic dimension of the paragraph due to the Svina Bromberg. And the theorem is that the asymptotic dimension of the mapping class group of the genus closed, oriented, connected surface of genus G is at most quadratic in the genus. So, we improve from exponential to quadratic. So, and and this proof doesn't live in the mapping class group word. So, in order to prove this, we construct some spaces. So, kind of that are intermediate between the mapping class group and the curve complex. And those spaces, they are HHSs, but you know that's the structure that they have. They're not, they don't appear elsewhere. And so they have a distance formula. And so this machine, a bit of the machinery that we developed goes into, you know, constructing these spaces and showing reasonably easily that they are HHSs. So, even if you're, if you only care about the mapping class group, the proof of this theorem, and presumably the proof of other theorems in the future, we'll use, I mean, use hierarchical hyperbolic spaces that are not mapping class groups. Okay. That was the propaganda. Okay. And now let me get to the meat. I'm going to talk about boundaries. Yeah. So, instead of the, ah, I'm just elated. So, instead of the distance formula, there's a, so, right, so notice that a consequence of the distance formula is that if you have two points that are, that have closed projections in every curve complex, and they're closed. So that's what you, that's in the axioms, but that's actually much, much easier to prove than the distance formula. It's like one page. Boundaries, right? So this class of spaces generalizes hyperbolic spaces and the boundary of hyperbolic space is very useful. So it's natural to try to find the boundary for an HHS. So it's very easy to describe it as a set. The topology is more complicated. So I won't write it down here. But as a set, it's really, really easy. So you have, so let's say that you have X with its collection of hyperbolic spaces and this is an HHS. Then the, the boundary of X with respect to this HHS structure is the set of formal barycentric sums, AP, sorry, AYPY. So it is a barycentric sum, AY greater or equal than zero, some Y is one of what? Let's say some for Y in F. So and F is of pairwise orthogonal elements of S, of disjoint subsurface in the case of the mapping class group. Okay, so why, why this? Oh, sorry, sorry. And I didn't say where PY is. It's important. So and PY is in the boundary of CY. So that's a hyperbolic, I mean, that's a hyperbolic space. It has a gram of boundary. Okay, so why this definition? Right, so sum of AYPY. So it is a barycentric sum. So and right, so it's going to be automatically finite. So it's always going to be a finite sub. And right, so you're summing points in the boundary of pairwise orthogonal domains. No, no, so, so any two elements of, of Y, any two distinct elements of F are orthogonal. Let me draw a picture, maybe. You have to choose some subsurfaces. So you choose subsurface Y1 and subsurface, I don't know, Y2 and maybe an annulus or something, Y3. So these have to be disjoint, orthogonal in general. And you're summing points in the curve graph of this, this and this. Yeah, it varies over all Fs. So it's the sum of these when, you know, you vary the very centric subdivision and the set of pairwise orthogonal things. Okay, so why this definition? So for every set of pairwise orthogonal elements of S, so you should think these joint subsurfaces, there is a product region. And now you're going to infinity in that product region. So you're thinking that you're going to infinity in that product region. And so you're going to infinity in one of the, let's say in both, let's say in both curve graphs. And so you have a point in one curve graph and a point in the other curve. And the slope determines the coefficients. So this is why this formula makes sense. I mean, this is why, how we come up with this. And so I won't say anything about the topology. So let me tell you about basic properties of this. And the theorem with Matt Durham and Mark Hagen is the following. So if we take an HHS and we assume that X is a proper metric space. So that holds whenever it's a calligraph, which is most often or posizometric to a calligraph. Then, first of all, you got a compactification. So the HHS closure, which is X union its boundary, is compact, house dwarf, and X is dense in X bar. So all the things you may want. And essentially, when you're working with the boundary, you don't need to know the topology, you just need to know that the boundaries compact and a couple more things. Okay, so here the hard part is compact. So the other properties are very easy, compactness require quite some work. But I didn't tell you what the topology is, so it doesn't matter. So this, as I said, hyperbolic spaces are HHSs. So is this the same as the ground of boundary in that case? And the answer is yes. So when X is hyperbolic, then the HHS boundary is, well let me write it slightly informally, is the same as the ground of boundary. And this statement is independent of the HHS structure. So there are different HHS structures that you can put in the same hyperbolic space, which if you ever thought about relative hyperbolic you shouldn't surprise you at all. And for, I mean, independently of the choice, you get the ground of boundary back. Right, okay, let me write it maybe here. So let's take maybe G hyperbolic, but also hyperbolic relative to a subgroup. Okay, then, so from the fact that it's hyperbolic you have the HHS structure that only contains the calligraph of G. Just one element. So from here instead you get the HHS structure where you have the condom graph and calligraphs of the cosets. Let me write it. So one copy of the calligraph of H for each coset. In this case there's no, there's no orthogonality. And so these sums are just made of one element. So this says that the boundary of G is the union of the boundary of the condom graph, union copies of the boundaries of H. I mean, that's it. Okay, so for, let me tell you one thing here. So for x, a cut zero cube complex and then the HHS boundary is, is marked against a simplicial boundary. Okay. These are the basic properties of this compactification. Okay, now I have to decide what to, okay, let me very, very quickly tell you. So there are various things that we do with the boundary. Let me just mention kind of as a teaser and then you can ask me more after the talk if you're interested. So there's a, the HHS boundary can be used sometimes to show existence of rank one element. So the suitable generalization of pseudonozovs. And so this, so from this we, we get back, when we specialize whatever theorem we have there, we get back handleMosher's omnibus subgroup theorem. So which says which says that if you have a subgroup, a finally generated subgroup of the mapping class group, then it either contains pseudonozov or virtually fixes a subsurface. Thank you. And also Capras gives rank rigidity for cut zero cube complexes. So in, for all of those that we can prove that are cut zero. Okay, but right. So we have the most annoying, the biggest drawback of the theory up to, at this point is that we cannot yet prove that for every cube complex that admits a proper co-compact action. So that every cube compass that meets a proper co-compact action is an HHS. So, which should be true. And so one way of proving it, so you can have fun with this. So, so please prove it. Please, please someone prove it. We're getting in very annoying. So if you take, so X gets your cube complex. When we're proving it would be to show the fault that the answer to this question is yes. So you take a cut zero cube complex and suppose that there exists a group G acting on X properly and co-compactly. So geometric. There exists some convex embedding of X into the universal cover of a Solvetti complex. So that's a non-equivarian specialness. So, some think it's reasonable, some think it's not. But yeah, I know. Okay. Convex, even just to be convex. Yeah, yeah, yeah. That would be just one way of, I mean, so if the answer to, yeah, so if the answer to the question is yes, then you can prove it. So the reason this question came up is that because this is exactly how we show that the complex special cube complex is RHHS. I mean, yeah, but in that case you do it equivariately. So this is non-equivariate. Okay. Now the last part of the talk, 10 minutes, going up under the following question. What is a geometrically finite subgroup of the mapping class group? Okay. We're in general finite HHS, but let's stick to the mapping class group. So, well, so in the paper we have a candidate definition, but okay, my quarters feel more strongly than me that that's the correct definition. So I'm not. So, okay, so, but, so whatever that is, so the following should be geometrically finite. So the following are examples. So whatever definition you come up with, if it excludes one of those examples, it's the wrong one. Following ought to be example. So one, so convex co-compact subgroups. So one possible characterization of this is that the map from G to CS is a quasi-zometric embedding. So these I will not define. So each subgroups, finitely generated which subgroups, I guess there are a couple of different ways in which this word is used in the literature. So in this case you have a nice hyperbolic plane intact mirror space and this weak subgroup preserves it. So that, you know, that really feels geometrically finite. And so there's a lining error read and have a way of combining different rich subgroups which is anger reads combinations. It's not going into not going in there, but it's very, very similar to something you can do in hyperbolic space. And, you know, it really should work. I mean, it should really give a geometrically finite subgroup. Okay, so compare, masquade combination theorem. And this, right, so this is interesting because they can produce examples of subgroups of the mapping class group that are as close as possible to being non-suit, from purely pseudonozo, without being purely pseudonozo. So this is to say that every non-suit and non-suit element in the subgroups that they contain is conjugate to a power of a fixed end twist. So there's the least possible amount of, the least possible non-trivial amount of non-suit and non-suit elements in there. Okay. And so why am I telling you this? Because a geometrically finite subgroup should have an extension to the boundary. So the inclusion of a geometrically finite subgroup, let me say need, need to have a boundary extension for it to be a proper definition. Because I didn't mention it before, but the natural question is how does PML compare? So you can consider PML as a compactification of the mapping class group. So if, you know, how does, so is this any better or worse than PML? So, and I want to argue that for the, for the purposes of having a theory of geometrically finite subgroups, our boundary seems to be better. Same as that one. That one, two, and three have boundary extensions for DHHS boundary. So, right, these are all hyperbolic groups. So the extension comes from, goes from the hyperbolic, I mean, the ground of boundary to DHHS boundary. Okay. And it's not true for PML. We bumped into a compactification of the mapping class group that it's not true here due to work of Leninger. I mean, not in general here for the work of Leninger. Right. I think this has evidence that we are on the right way. Like we are doing something that even for the mapping class group is better than what was done before. Okay. Unfortunately, I still have five minutes, which means that I'm going to have to give us a catch of this. So very, very rough sketch for Leninger, for VH and Leninger read. So what are these VH subgroups? So what they do is they take some VH subgroups, AMB, the mapping class group of the genus G surface. And these are isomorphic better free groups, but you should think of them as fundamental groups of puncture, the once puncture surface of genus G. And there, the cast subgroups coincides. And I'm augmenting to surfaces with boundary along the boundary. Okay. So this goes back to Moons question actually. So the cast subgroups. So the, so A is hyperbolic relative to H. So that's just for A. So which means that you have, so you have the cone of graph and the cosets of H, of this cyclic subgroup. And this has a, I mean, the map, the inclusion map induces a quasi-zometric embedding of the cone of graph into the curve graph of CS. So in particular, there's a boundary extension here. Okay. So from the boundary of the, the cone of graph to the boundary of CS, as a, which you can think of as a subset of the HHS boundary. And GH, these are multi-twists. So it's also easy to find a boundary extension for those, because you're just moving in a then twist flat at a, with a constant slope. Okay. And so somehow the, the whole point. So, so it boils, it all boils down to, to showing that the following picture is correct. So you have the map from the amalgamation, H star B into, into the map, well maps to the mapping quest group. So this, so this guy is hyperbolic relative to H. Yeah. It has a cone of graph. And this should still map to CS in a way compatible with that embedding, you know, quasi-zomatically embedded way, which is the case, which happens to be the case. And well, I can stop here. Thank you.