 Thanks very much, and thanks very much to the organizers for Inviting me here. I always enjoy visiting Italy though. I've never been to this part before It's nice. So okay, so let me start out by talking about Relatively hyperbolic Dane filling and we'll use that to kind of introduce some of the objects that I'm going to talk about Maybe before I start writing I'll just say the cannon conjecture is is not on the face of it anything to do with Dane filling It's about hyperbolic groups with a two-sphere boundary, and you all know examples of hyperbolic groups with a two-sphere boundary Fundamental groups of closed hyperbolic three manifolds and the cannon conjecture says that's it okay, so All right You can make a version of that as I said in the abstract from for relatively hyperbolic groups And so part of what I'm going to say is about what what those are and how you might connect the two families of groups So we'll start with Classical Hyperbolic Dane filling so this is this is this idea of Thurston so you start with M Hyperbolic Orienable finite volume with some cusps So for example You just might think about s3 Minus the figure eight not which I don't know if I can draw it quite as fast as In did but there you go So the picture you have is of course not of this you're thinking about the complement The geometry is there's some sort of fat topology in the middle and then There's these cusps. So it's something like a hyperbolic surface so Except the cusps Have a tourist cross-section Instead of Instead of a circle They go on to you know to infinity. So there's some cusps. So cusps See that's the red stuff And now Maybe I don't like non-compact manifold so I can drill out the cusps I can cut out the cusps and I can glue in some solid Torai so So we can choose Curves on boundary VAM sort of one for each The I so this is maybe this is C1 C3 And then we're going to so attach Solid Torai to M minus C So the meridians are glued to the chosen curves. So let's call them gamma one So the theorem so surgery theorem of Thurston Says that for most choices of the most choices of these gamma eyes I got a closed hyperbolic manifold and most you can you can be very Quantitative about what that means in both of the situations before I did the filling and after I did the filling There was a natural kind of boundary at infinity associated and in both cases Weirdly enough it was a two-sphere so So call this call the closed manifold M of gamma one up through gamma N and let's call Let's write just G for Pi one of M and G bar for Pi one of this thing, which is really G mod out by the normal closure of the gamma eyes so and G has some peripheral structure P Which is the set of the fundamental groups of the cusps. So purely from group theory We can reconstruct a sort of boundary at infinity So this group G bar acts in a nice way on hyperbolic space and therefore on the boundary and so does G But G acts in a little bit more particular way and that these groups in P are fixing points at infinity So they're bigger than the kind of groups that you would see fixing a point at infinity of a hyperbolic group, so So both Have a boundary at infinity which can be constructed from group can be recovered from group theory This group G bar is an example of a Hyperbolic group and this pair here This is an example of a hyperbolic group pair or relatively hyperbolic pair So other examples, and I'll just say examples rather than really give definitions So are if I have any kind of geometrically finite I Have gamma equal to geometrically finite Clinian group and here I mean of any dimension So maybe I'll put Clinian in quotes. I don't really necessarily mean a subgroup of ps l2c So just in s o and one And if I take P to be a collection of maximal parabolic, you know up to conjugacy Then I get then I get an example of a pair Is a is relatively hyperbolic Group pair and if this set of maxful parabolic is empty then it's just a hyperbolic group And these these are examples and the questions. I'm going to talk about our I think open even for that class of groups Just discrete subgroups of s o and one So the cannon conjecture Says that if G is a hyperbolic group With two sphere boundary then this implies That G is And here really I mean After killing a finite normal subgroup and maybe passing to a finite index subgroup. So it's virtually Clinian, so it's virtually a hyperbolic three-manifold group close hyperbolic three-manifold group So what I'm going to call the relative cannon conjecture today Is that there are many ways you can kind of relativize the cannon conjecture But what I want to talk about today is where you take G P to be a relatively hyperbolic pair Where the elements of P are abelian subgroups. So in general for a relatively hyperbolic group There's no requirement that these subgroups be anything at all. So they could be quite wild groups But today I want to focus on abelian parabolics. So say I have GP Relatively hyperbolic Well, there's a nice proper space that you can associate to any relatively hyperbolic pair And It has a Gromov boundary and I'm going to call that the boundary of the pair GP and Suppose that it's homeomorphic to s2 and then it's the same conclusion. So Supposed to be that G is Virtually Clinian. Yes. Thank you We'll totally hyperbolic With P a collection of abelian subgroups It ends up following that they're ranked two if you assume that the boundaries a two sphere So so you could say rank two abelian subgroups if you want Any other questions about what I said so far? but maybe I should say just a word or two about This I kind of said something in words, but they say it just a little bit more so what is Boundary of GP and here the answer is what I mean is the Bodich boundary if I have a relatively hyperbolic group pair then I can associate a Gromov hyperbolic space to it in the following way I start with the Cayley graph and then I equivalently attach Horribles to cosets of the peripheral groups so so this is equal to The boundary of a space. I'll just call x of GP And what is that so x of Gp is equal to the Cayley graph union Horribles in the case we're talking about today Where all my peripheral groups are ranked to abelian? I can think about literal hyperbolic horribles and just attach them somehow to the cosets Right in general I need to take the Cayley graphs of the subgroups and do some kind of exponentially decaying product With a zero to infinity and it doesn't really matter Sort of the details of how I do that as long as it as long as the metrics are exponentially decaying the same way the Metrics on horror Horror spheres are sort of exponentially decaying as you lift them up in the upper half space model The rate of exponential decay doesn't matter All those spaces that you build that way will be quasi isometric to one another So that's basically a theorem of of graph Whose work was mentioned just the other day? Okay, so that's what this boundary is and what what you get if you start with Something like this fundamental group of M relative to the fundamental groups of these cusps is you end up recovering something That's quasi isometric to h3 So one version of this was studied by cannon and Cooper some time ago Showing that when they showed that the class of hyperbolic Well, anyway finite volume hyperbolic manifold groups three manifold groups is is quasi isometrically rigid Okay Okay, so so we have this this general more general class here and the idea is that that we can relate these two classes By a generalization of this dain filling construction Can relate hyperbolic? relatively hyperbolic groups Via Dain, let me say what I mean by that so if I have So if I have G P a Relatively hyperbolic pair this collection by the way is always going to be a finite collection and if I choose A family of subgroups And so this is going to be a set of sort of and I which are normal in the PI where PI is in this curly P so then If you look at the group I'll write G bar Equal to and we might if we need to specify write G of n and What this is this is just G? mod out by the normal closure of the union of this collection and Normal closure in G This is a dain filling. This is a dain filling of G of the pair Gp That's the definition And you see that it kind of generalizes the group theoretic side of what was going on here because each of these gamma I so here I killed elements and not subgroups, but it's the same same difference, right? So in each of these cases, I'm killing a cyclic group which is generated by the gamma gamma I All right, and there's a group theoretic Dain filling theorem which is analogous to this theorem of Thurston and the form of this data is due to Dennis Ozen HDF theorem there are weaker versions by me and Groves and much stronger versions by Damani Juridel and Ozen But so what I want to say here is that so for any Let me make another definition here. We can let P bar be the set of images. So image of PI mod ni in G bar Right, so So we suppose P is relatively hyperbolic and Then there is a bad set This is sort of the set of exceptional failings the finite set. I should that's important my night bad set B sitting inside G so that for any Collection N as I just was describing with so this is B and G minus the identity So any N with the union of N Missing B Then you have a number of consequences. So one is that if I if you look at the pair G bar P bar is relatively hyperbolic second is That if you look at each of the elements of P bar so the for the image of PI is Equal to is well, it's naturally isomorphic to PI mod ni and then so let me put this in a box I don't want to Quite get rid of it yet. So more over for any Finite set in G there is a maybe bigger bad set B sub F so that if I avoid that set then F embeds into G bar So this is saying that I can exhaust Or rather I can do the opposite of exhaust if I if I am As long as I choose, you know, right? So if I choose a big enough bad set I can sort of get any finite Amount of data in G to survive Now what does it have to do with the surgery theorem? Why is this a kind of generalization of the surgery theorem? Well, the main point is something I haven't mentioned yet, which is that if you have a relatively hyperbolic group pair Where the peripheral groups are themselves hyperbolic Then the G bar is going to be hyperbolic as well. So maybe I'll just write that here. So if I have G comma P relatively hyperbolic and all In curly P our hyperbolic for instance virtually cyclic then G is Also hyperbolic And so what's happening in the original Dane surgery theorem is that the P mod ends are all cyclic groups And so I can actually I actually don't need them in the peripheral structure I can discard them get a smaller peripheral structure in fact an empty peripheral structure With respect to which the fundamental groups the manifold is still relatively hyperbolic. So it's actually hyperbolic, right? Okay so what The so what is the following so Well, we're able to show Well, first of all, let me just ask a general question Which is so in the classical setting we we have this wonderful situation Where before I didn't fill my boundaries a two-sphere after I didn't fill my boundaries a two-sphere in general It's not entirely clear what happens. So It looks like what happens is that usually things get a lot more complicated than that Let me just pose it as a general question so for Long Dane fillings so that when I say when something like this happens when there's a bad set like this We say that the the conclusions hold for all sufficiently long Dane filling right, so Maybe I'll just say that write that in another color. So This means these hold sufficiently long and and the same thing with this F embedding in G It's just what sufficiently long there means depends on the set F so for long Dane fillings is the boundary of bar related to the boundary of Gp The word red here, and I'm going to explain what I mean Well, firstly, I mean you can ask it for both. So the red means reduced So if I have a peripheral structure, I can get a new peripheral structure by throwing away all the hyperbolic groups in the peripheral structure Right, and that's a kind of a natural thing to do Bar red means Bar without hyperbolic things Sort of a partial answer. So there's some examples Really, all we have is Examples, well, there's a few things that we can say and I'll say say some of them. So the examples of Mosher and Sigev and also by Koji Fujiwara and myself In a setting where you can actually get something which is a little bit better than hyperbolic you can get a Space that's that's cap minus one that's being acted on so These examples suggest that usually the boundary gets a lot more complicated, but it has something to do with the old boundary So that's usually but I'm going to say it So usually the boundary of G bar P bar Reds say is Much more complicated. I'll say without writing down What happens in those examples so what happens in those examples is you start with sort of the nicest possible generalization of Hyperbolic three manifold of finite volume you talk about a hyperbolic n manifold of finite volume right and now your peripheral groups are z to the n minus one and In the Mosher Sigev examples, you just kill the entire parabolic and in the ones that I study with Koji Fujiwara, we kill like a code of co-rank one subgroup, right? So everything except a cyclic group both of those situations your peripheral groups are now either the trivial group or Cyclic group so you get something hyperbolic so it makes sense to talk about the boundary and We show that the Czech homology of the boundary so it's you start out with a sphere and then you end up with something whose Czech homology is infinite dimensional in lots of dimensions, so So it's much more complicated than a sphere. So you don't usually get a sphere And maybe there's yeah, so there's something special about this low dimension all right Okay, so here's something you can say is a theorem with Daniel Groves all this theorem One because I want to refer to it later Which is that so this is Groves So let's just let's assume let's assume from now on that P consists of abelian groups. It's true in more generality but That's plenty for what we want to do today so One way to say what we said is what we proved is that if And this is true in more generality. So if GP has No elementary splittings is true P bar P bar, you know for long fillings So using work of Bodic in Grof This means that in particular Have the boundary being and this is pure red Well, it doesn't matter Okay, so if I have boundary of GP connected Without local cut points This implies that if I look at boundary of G bar Say P bar red is also connected without local cut points The reason for that is that connectedness has to do with free splittings And cut points have to do with splittings either over parabolic groups so cut points have to do with splittings over parabolic groups and Cut pairs if you have a local cut point You've usually got a nice cut pair that's going to give you a cyclic splitting So so that's the connection between those things. So we can say a little bit about the topology at infinity No, it's not a three-dimensional statement. This is just a statement in general Yeah, an abelian is much stronger than what we need for the so to get the statements about the boundary We do need to some assumptions because these theorems of Bodich and graphs that we're applying require Yeah, and I'm being a little bit sloppy about what I'm saying too. Anyway, you can you can ask me about particulars later If you have a particular example, you you care about. Oh, right. So what I want to talk about right, so now I've introduced what I'm talking about and Let me just say the two kind of mean statements. I want to talk about today. So the theorem a is That Well, okay, so I'll just state the one that's in the abstract first So it says that if the the the cannon conjecture Implies the relative cannon conjecture. Maybe you think the cannon conjecture is false But you should still be happy about this theorem, right because it means that it's easier to find a counter example Because you can go to this world of relatively hyperbolic groups Okay, but there's this comes from an absolute theorem that doesn't have to do with relating different conjectures So that's theorem B Which says that? So if you have So let's say we have G P A relatively hyperbolic pair with Boundary equal to the two sphere and P consisting of Abelian subgroups, I would guess this isn't necessary Well, I'm just gonna write down what we proved What it occurred to me I could prove this week maybe So if this is true, then For all efficiently long Classical feelings, okay, what's a classical feeling? I'll tell you that in just a second, but it's something analogous to what happened for a hyperbolic three-manifold then the boundary of G bar Well, first of all G bar is a hyperbolic group. So I can just talk about the boundary of G bar and What you'd like to say is so a classical feeling is where you kill an infinite cyclic subgroup of each parabolic, right? And so what you want to say and what would be true if they if If this if we started with a clining group is that the boundary of the result would be a two sphere all the time, right? So we don't know that but we can show it's either a two-sphere or a subpinskied carpet and that's good enough It turns out you get theorem a Maybe just just write this Classical this means that each Ni is So I'm either doing a If it was a three-manifold I either be doing it ordinary dain filling or an orbithold dain filling Any questions before I? Talk about why theorem B and plus there may Yeah No, no, it's the relative. Well, it's not compatible. Yeah, that's right. It's not so if either if the conjecture is true Then you you actually never get the sypinski carpet. That's right Yeah So, I mean I should say like how do sypinski carpet? Boundaries appear in nature And not in imaginary counter example land So if you take a hyperbolic three-manifold with totally G desic boundary and No parabolic right no cusps and look at its limit set. It's going to be a sypinski carpet, right? so the the the Elevations of the boundary components are giving you these hyperbolic planes and the whole limit sets on one side of each of those So you're getting s2 minus a bunch of bunch of disjoint discs Discs with disjoint closures. I should say I mean I'm not sure you could conjecture it in general. I mean there's I don't know any counter examples in the theorem Yes, in the theorem. I mean this version with p a collection of abelian subgroups. Yes Okay The sypinski carpet situation so we have to deal with that somehow, right? So if I've got my filling So what I'm going to do is I'm going to get a bunch of Clinian groups if it's all s2 then I just automatically get my quotients being a bunch of clinian groups from Canon conjecture, but in fact I also get that for sypinski carpet boundary, too And that's by a theorem of capovitch and Kleiner so the Canon conjecture Implies that Let me use maybe a different letter any hyperbolic Gamma with boundary of gamma a sypinski carpet is You know virtually Clinian So for now, let's just ignore all the virtually stuff So what you what you can So for the purposes of explaining something so ignore virtually and Start with your relatively hyperbolic group pair. So so we assume that I have a relatively hyperbolic group pair with boundary s2 And now choose a kind of co-final sequence of classical fillings so GI bar So what do I mean co-final? I just mean that If I look at any, you know, if I look at any finite set then it's going to embed in all but finally many of these things so the intersection of the kernels is is trivial and so Capovitch and Kleiner together with the Canon conjecture so CC plus KK Implies that for each of these you get you get an embedding and forgetting the virtually into SL2C so bar embeds in PSL2C discreetly so now I can think of each of these as a Representation of the original G and so I got a sequence of points on the character variety of G sequence of Representing representations a stably faithful sequence of representations And Stably faithful and now if I think about the characters Well, there's either convergent sub sequence or there isn't if there's a convergent sub sequence Well, then they're going to converge to Representation of G that is faithful, but you can also show is discreet so I Guess maybe it's not you prove them both at the same time so we can show that if Chi I converge to Chi infinity and Chi infinity is treat Faithful and I get to stop So if on the other hand the characters diverge Then I'm getting this Because they're stably faithful and they're diverging well some traces are going to infinity So I mean well, I guess that's what I mean by diverge Anyway, you can rescale the action on hyperbolic space And in the limit so this is a kind of standard Technique and geometric group theory in the limit you get an action on an archery and you can show that that's a stable action with a billion arc stabilizers, so Otherwise The Chi I converge to stable action Archery with A million arc stabilizers, okay, and then we apply the rips machine which implies that The original group G Has a splitting over an abelian group and so that's going to have to be An elementary splitting I think of a GP has an elementary splitting which implies that the boundary of GP Wasn't a two-steer Because it would have to either be disconnected or have a local top point, right which S2 doesn't all right So that last part of course is using Well, no, it's well. No, it's not it's just using what I just said. Sorry, okay but we another way to think of it is that you have a contradiction to the statement that Actually, sorry. I'm just I'm tying myself in knots here Let me not say anything anymore about that and let me move on to the next to prove to the sketching theorem So the idea here is that we want to use So we're going to use a criterion of clay tour To show that the boundary of a long filling is planar Right and so we're allowed to use clay tours criterion because we already know That the boundary doesn't have any local cut points so So the theorem one Implies that if I look at so from now on G bar Is just going to be some very long filling No cut points, so there's a theorem of clay tour does that Yano continuum Now cut points if and only if It contains no top a lot obvious topological obstruction of being planar. So so it's the analog of Kurtowski's theorem Same as Kurtowski's theorem. It's just that you're allowing a continuum and not just a graph So What so we want to So we need to try to show That no such graph so no K 5 or K 3 3 embeds And how do we do that? Well, we need some way of approximating the boundary and Idea is we're going to approximate boundary of G bar by a partly truncated quotient Sorry by the boundary of a partly truncated quotient of Of the space the model space for GP of the ZN spaces where ZN is a partly truncated quotient X of GP by a finitely generated free factor of the kernel of the map from G to G bar So it's a result of the money good on Ozen that for long fillings the kernel of the map from G to G bar is freely generated by some collection of intersections with parabolic right and so The way that's proved they sort of like they they get an exhaustion of it by sort of infallibly many factors at a time And that kind of doesn't work for our argument because we want we want finitely generated free factors So we kind of give a new argument for that same theorem But so let me just point that out. So DGO Says that this kernel let's call it K bar again for a sufficiently long filling And in this case, it's a classical filling. So it's a free free product of cyclic groups. So it's freely generated We give another description, which is which is basically the same. So except sort of filtering it more slowly the Austrian of K by finitely generated quasi-convex, of course not uniformly quasi-convex Free subgroups of free factors. Well, I'll call them sort of F sub n And so the idea is that so f sub n Let me just draw Kind of picture here one dimension down So this is a sort of picture of h2 and So I've got Say two If I look at the subgroup of my filling kernel that's generated by say parabolic fixing just two of the points I can sort of delete horror balls around those points but leave alone the rest of the points and now look at So the elements of the kernel are sort of moving everything very very far along these horospheres. So so my element fixing this Pointed infinity is going to send this maybe to this over here really they're going to be so vanishingly small You can't see them, but I want to draw them a little bit visible This is going to get identified to this and so what we sort of end up with is This Picture like this. Let me let me just draw it flat. So if you think about what the quotient There is I've got this sort of fundamental domain for it in the middle sort of the orange part It goes out Infinity Okay, another way you can think if I didn't delete the horror balls It would be I just a sort of attach a couple of horns to this picture right sort of going down these cusps going down Now this space is not simply connected, but it is delta hyperbolic and By choosing the depths of these horror balls carefully we can show that these spaces are uniformly delta hyperbolic So I have these uniformly delta hyperbolic spaces where I've sort of taken the boundary of the original space and Quotient it out by bigger and bigger subgroups of the kernel K And it's these spaces those quotients are which are going to be the approximations for the final boundary So so that's what I mean here by a partly truncated quotient of X by this finitely generated free group So so we show That the boundaries of these ZNs Converge in some sense To the boundary of G bar and in fact we need to be a little bit We need to have a fairly strong statement. So So we show in particular there are Visual metrics on these spaces so that so and Sort of K well not K lambda epsilon sub n quasi isometries from boundary of ZN the boundary of G bar with Lambda fixed and epsilon n Going to zero So there's sort of we can't make them isometries, but they get closer and closer to kind of by Lipschitz maps right and under these conditions So we There's an argument of Ivanov. Oh, right more over We show that these things are spheres So that's sort of obvious in this example that I get a one sphere at infinity But in general in fact we also get we get two spheres in our setting. So also the boundary of ZN are Spears With uniformly linearly connected metrics Okay, I didn't know what that meant Before I started this project Okay, I'm going up because I'm getting near the bad part All right, so let me just say one more sentence and then and then I'll stop So the final thing that we do is that we apply this argument of Ivanov Which was used in a different context Which shows that under these circumstances he had some slightly different circumstances, but the argument still goes through that any graph that I could embed in This boundary I can embed in all but fundamentally many of these boundaries so So if a graph embeds in Boundary of G bar and embeds in All the finitely many boundary of ZNs All right, so this gets us that Using clay tours criteria and this gets us that the boundary of G bar is planar And then the fact that there is no local cut point Together with another theorem of Cavavich and Kleiner gets us that in fact It's a it's a Sapinski carpet or a two-sphere and that those are the only possibilities and so that's the end of the argument I'll stop