 Okay, so I do want to clear up a couple people ask me that convex hole in general is not convex So just so you know That it's also called some places in the literature the weak convex hole, so Don't It just want to clear that up Okay, so let's do some examples. So I want to talk about bodice is splitting so Yes, it is quasi convex. I might have said that but I called it the convex hole Okay So I want to talk about bodice. So what is bodice that you can see in the boundary? so remember you can see the boundary of quasi convex subgroups and so bodice is going to see a lot of quasi convex subgroups and How they are put together So let me just do an example Okay, so let's do an example Okay, so I'm going to start with a space and my group is going to be the Put them on a group of that space. So I'm going to take three surfaces With boundary I'm just giving them different genus so I can distinguish between them Okay, and I'm going to glue them so three surfaces all with one boundary component I'm going to glue them together along their boundary component. So I'm going to glue them all to this one Circle now this embeds in So I can embed this in our Three I can clearly embed when I glue two of them together and the third one you can put on the inside or the outside I'll just draw it on the outside Like this okay, so I have three circles you can think of I'm kind of going around swinging like this It's a little easier to think of the universal cover. Okay, so if I thicken this up in R3, I'm going to get a hyperbolic three manifold Hyperbolic with boundary. It won't have totally geodesic boundary Okay, and by Thurston because this is Hawken. I don't need to use Perlman This is realized by a geometrically finite, which is quasi convex on climbing groups I can realize this as the take the convex all of the limit set of this so this group is a subgroup of isometries of H3 and if I look at its limit set It'll be quasi convex in the structure So I can there's lots of different ways I could do this But there is one where I can look at the limit set Take the convex hole that'll be some three-dimensional thing the universal cover and the quotient will be a convex quasi convex manifold Okay, so you can think of this as a three manifold that's hyperbolic that's hyperbolic Good example of a hyperbolic group. So let's look at think about what its universal cover and what does its limit set look like? What does its boundary look like? so All right, so now This all of these so this admits a graph of groups decomposition or have like a four F2 and F6 corresponding to these Surfaces and they're all glued together along this z Okay, these z subgroups are quasi convex So I can think of I'm going to take that curve which I want to call a although I didn't label it And I can look at its universal Some elevation of it. I'm thinking of this in H3 Okay, so I'm going to look at So there's two different natural spaces on which the sex one is to get a A geometrically finite cloning group. So for that I'd have some three manifold. I'm thinking it thickened up I can also just think of this complex. It's a little bit easier Both of these are quasi isometric to the group and the complex is going to have these little pieces of hyperbolic um space so each Surface group is going to act on a convex subset of H2 let's just think about that for a second. So if I just have like say f2 Okay, so it's uh So here these are the lifts of this boundary curve. So here's a structure for of two Okay, so these boundary curves here that I've drawn are just lifts of this boundary curve right here so I'm taking a geodesic on the surface and Its boundary is going to be a canter set which is going to be naturally embedded and In this circle All right, so this is the space here's my space x on which this hyperbolic group one of the spaces x also There's a free group. So this is my space x on which this hyperbolic group acts geometrically And this quotient is just this nice surface with totally geodesic boundary All right, so those are going to appear so Bodishes says that these are actually also quasi convex subgroups and those boundaries are going to appear So they're going to be hooked up to the boundary of this a so the boundary of this z is two points So here i'm going to call this say maybe this I take some elevation Of um the curve a it's left invariant by the cyclic group I'm going to call g sub a this is going to roll up to give me that circle there and these um i'm going to have These little canter sets that are coming out here So i'll have this Little pieces. So this is like one of those So I take one of those curves here This is going to be glued Right to this curve up here. So this is like i'm starting to draw the universal cover of this complex And then i'll have another one coming out over here for the other one So i'll have this canter set On the boundary Sorry, I didn't follow so you take your Kind of flash to go well, you know two-dimensional picture here. So this is going to be one of the pieces and you glue Oh, yeah, and I'm going to glue three of them together along one of these lists Exactly. Okay, so the endears description is better than the drawing So each of these surface group acts on a convex subset of h2 So to start to build the universal cover. I'm going to take three of those Glued along their common lift. So that's where these three surfaces are glued together Okay, so i'm taking three of these and i'm trying to draw this picture here So let me draw the last one out here I'm going to have a canter set coming out here I have this last sheet back in the back Okay, so right up at the top I have three of them three sheets are going to be coming together to meet in one of these limit points That's stabilized by this Okay, and then i'm not going to draw it but out of each every one of these there's going to be another three sheets coming up Okay, so I can start to Okay, so um Let me um do some color here. So I just really have color. Yes, and I will say that again So here's my One of my sheets coming out Here is The orange another sheet And then i'll make a third sheet. Let's make this blue Okay, so there's three sheets coming out and then from this orange sheet. There'll be a green sheet and uh, blue sheet Okay, and all the orange sheets are going to map down to the orange surface I'll do this And then I'll take your question So I have like the orange surface all these are going to map down to say the orange surface the blue surface and the Okay question george Okay, so I just drew one elevation of this surface So every lift every elevation of this circle Is going to be a geodesic and each one is going to have three of these things going off from it Exactly that's exactly right Thank you. Sorry trying to keep he's valiantly trying to keep me from killing myself Okay, so so this is what this looks like so now I can take this Um picture maybe I'll just draw it again over here So I've got my curve a And I've got my orange surface coming out Blue surface coming out And I've got a green surface coming out And the limit sets that these quasi convex subgroups are going to The canter sets which I'll see in the boundary Is a these are all free groups their boundaries are all free groups Now I can form uh, um a graph By how these things are glued together by how I'm just thinking in the universal cover now Bodice can think in the boundary, but let's just think in the universal cover So I've got these elementary subgroups. So I've got these two-ended subgroups and whenever I see one of those I'm going to put a white, um A white vertex Okay, so the two-ended Stabilized by I just tuned a group. I'm going to put a white vertex And then if it's stabilized by a free group, I'll put a black vertex So sorry, what are you doing? You're building a graph? Yeah, mm-hmm So I'm looking at this universal cover in a minute. I'm going to just get this from the boundary But let me just it's easier to see the graph. I think if you think about this universal cover here So I've got these sheets I've got these lifts of a so I have all the elevations of this curve a here And those for each one of those I'm going to put a white vertex on each one of those Okay, then I have all the elevations of my surfaces That's that's going to form the whole universal cover because that's all there is And for each one of those I'm going to put a black vertex And each one of those Okay, and then I'm going to connect I'm going to connect The white vertices only this is going to be bipartite. So the white vertices Don't touch any of the other white vertices when they meet Okay, and so group theoretically I can say this subgroup the stabilizer of this subgroup Is actually there's a map between These the stabilizer of this subgroup contains some of the stabilizer of this subgroup Okay, in particular this z is a subgroup of that free group of all three of them So this z right here Is a subgroup of this group? It's a subgroup of this group and it's a subgroup of this group because it's just this boundary curve right there Okay, so I'm going to get this Trivalent graph Right here. So let me just draw the graph. So Right here It's going to look like these are the three so notice. I'm getting the splitting Okay, for I have these I'll just draw all these brown Okay, now what's going to happen here to this vertex? Okay, so this vertex maybe corresponds to this elevation of this orange surface, which is just a hyperbolic sheet Okay, and that's going to have infinitely many of these All of these boundary curves here. They all map down to the same boundary curve Okay, and they're all glued to this circle. Okay, so they're all going to be There's going to be infinitely many Like this white vertices And then each of those is going to have three. I won't draw that much more Each of those is going to have some black vertices going off of it Okay, so I'm starting to draw the best their tree of the splitting if you know what that is Okay, so the group is going to act on this graph. Okay, just because it acts on This universal cover Okay, but it's going to have Very non-trivial stabilizers So like the stabilizer of this vertex here is going to be z. It's the same as the stabilizer of that piece that I labeled it with Okay, the stabilizer of this little sheet right here is going to be whatever it was f free grip over rank four Okay, and same so I'm going to have all these stabilizers Going along and then these when the stabilizers intersect. I'm going to have these edges Okay, so that's how I can form that so you can see this graph Also, so boat it showed so let me just say what a splitting is Say what boat it shows Okay, I think I've had that up for two minutes, but maybe I'll write it over here So definition elementary splitting so you can think of an elementary splitting as like a graph of groups or you can think of it as This tree and you have these groups acting on the you have your group acting on the tree and then the Subgroups the stabilizer subgroups of vertices Give you the vertex groups So definition the elementary splitting Is a minimal So this just means it doesn't fix a proper. There's no proper invariant subtree uh co-finite action on a simplicial bipartite tree Okay, and I want my uh Tree is going to equal um The uh, I'm not going to call the tree because I'm going to use t maybe later. It's like Okay v e v in e and then my edges Okay, what does this e mean so e stands for elementary? So these are the elementary vertices. So these are the ones that are stabilized by two ended groups These sub e vertices are stabilized by and I will sometimes Confuse a vertex which is stabilizer. So I'll talk about that that vertex And I mean if I'm talking group theoretically, that's the stabilizer and the v and ne vertices Or I have some other stabilizers Not too ended Okay, so that's how I'm thinking of a splitting So if you like thinking of it as a finite graph You can just take the quotient by the group and you'll get the graph of groups decomposition that you might be more useful used to Okay, so um these the Vertices so bodice shows I wouldn't bodice do so bodice um Give him in a group that was one ended So gamma one ended um it's hyperbolic And it's not um, I don't want to deal with um Virtually focusing groups. Okay, just because there's a just a little funny example where it doesn't split So it doesn't have enough structure. It's too homogeneous Then you can look at so he devised a way to just look at the boundary to get Look at the boundary to get uh elementary splitting It'll actually be maximal and canonical, but let's just think how we get an elementary splitting Okay, and for this example right here the splitting that he gets is exactly this one this tree Um with this action on the tree So let's um talk about how he does that from the limit set Or the boundary So how do you look at the topology of the boundary? To give this so you'll notice so those two points that are the end points of that elevation of a so that long white geodesic When I um remove that whole geodesic It's going to split the universal cover into three pieces Okay, so and if I if you look at those two points on the boundary when I remove those two points on the boundary It's going to split the boundary into three pieces because I've got these three disjoint sheets going off to infinity Okay, so that's called a cut pair And so what if you look in this boundary you can go around you can see all these cut pairs And those are all going to correspond to these elementary splittings So let me just I need to say a little bit of Let's look at the local cut points Okay, so the valence um, so I'll just call that Okay, so the valence of x where x is a local cut so what does it mean to be a local cut point? It means that when I remove it so that means um Minus this point x has more than one end Okay, so the valence of a local cut point equals just the number of ends Of um Okay, so he's going to take all the set of local cut points So bodice puts their equivalence relation on all the set of local cut points so this example Although in some sense the simplest one you can do is sort of complicated because Everything there's local cut points It's sort of like all local cut points because I'm going to have these the boundaries of my Um of my candor sets are also going to be local cut points Okay, so uh, so I'm going to say that x is equivalent to y if the valence of x is equal to the valence of y and um x y is a cut pair Okay, so for example the two points at the end points of that geodesic upstairs are going to we're going to have this So each has valence three And when I remove both of them, I'm going to get three pieces. Yeah when you when you say one more than one the end you mean the end you mean the connected components Yes, you take a compact exhaustion and you want to say there's always going to be so many Not not connected components just end as like a topological space So when I remove just one of those when I remove both of them, that's a cut pair But if I remove one it's actually not going to separate the boundary into other components But that space is going to have three ends if I start looking at bigger and bigger compact sets I'm going to end up with like three ways to go to infinity. Basically So just when I just remove one it's not it's not a cut point. It's a local cut point So if I just remove one it's not going to separate the boundary But if we remove both the pair that'll be a cut pair So being a cut pair means that if I take the boundary minus these two points and there's more than one component Okay, um so He looked at these equi so these equivalence classes So the equivalence classes So um when the size is two They're going to correspond to they're going to be stabilized by so if the size of an equivalence class Um is equal to two the stabilizer of that equivalence class is z and um These correspond to the elementary vertices So that's sort of the first thing to see And I can also have valence two um So the the cantor sets valence two with infinitely many equivalent points and uh the closure of such an equivalence class closure of equivalence class is going to be a cantor set So this is what I have In this example, and I'll write this up more formally in just a second, but I want you to see So if I remove two maybe that one's harder to see So if I remove those end points of that g Elevation of g you can clearly see those three sheets coming out But if I remove the cantor set, what's going to happen? Let me draw a little Magnification of what's going on at the cantor set level If I have uh say an orange cantor set It's hard to draw the whole cantor set And then I'll have like a blue one coming out of it and a green one coming out of it I'll actually have a whole green one coming out over here And then on the other side I'll have this is where I started to glue the other sheets and then I never showed you Yes, I can thank you Okay, uh This one I'm going to write that out more formally in a minute. I just want to give you an idea Okay, so what's going to happen is like two points like this On these cantor sets that aren't jump points So the jump points are what you kind of see it's going to kind of divide this Whole thing the whole limit set. There's another blue one coming through here So I'm just trying to draw the boundary there It's going to divide this boundary into the piece in between here on this This is sort of looks from far away like a circle. That's actually kind of accurate here That divide that into this piece right here and then this other whole other mess Okay, so these are going to be all these the non jump points of the cantor set are going to be these Are going to be valence too and the closure of that class is going to be this whole orange Cantor set And that is going to be the boundary of a free group. Okay, it's a boundary of a free group and it has Let me just write all this down It's called a cyclically hanging Free group, which I thought was a terrible word until I started to think about it. And now I think it's great Because it's cyclically ordered and it hangs So let me just write what bodich Said here. I'll use just boards. I'm trying to use all my boards So there's three types of this. Here's the types of equivalence classes I'm just describing this. I'm not proving any of this Bodich's paper on Splittings and cut points is very good. There's also lots of Lots of expositions. I have a survey article with sam kim Which I will put up on the archive or I can send it to if you want it has exactly this example And some other examples which I might get to okay, so the types of equivalence classes So I could have the valence of x could be three or more And the number of x is going to be equal to two So you can't do that unless it's just two unless it's z So whenever you have a pair of local cut points And you have In its valence is three Then you're going to have and that's a cut pair then it's going to stabilize a two-ended group so these Are stabilized by two-ended groups. So notice i'm kind of Going back and forth between the cut pair And the vertex because the cut pair is going to correspond to a vertex One of those vertices one of those white vertices up there. So those are going to be like white vertices Soon correspond to white vertices Okay, I could have the valence of x could be two And the number of x just equals two still Okay, so that's also going to be a white vertex And then I could have the valence sorry valence Of x be equal to two and the number in the equivalence class be infinite Okay, when I have such an equivalence class like this, this is going to correspond to a cyclically hanging um fuchsian group so there's a So I take the closure of that equivalence class. I should do it like that And this is going to map into so this is going to be a canter set And it's going to map into a circle And that the stabilizer is going to preserve the cyclic ordering Okay, so the action on this equivalence class Is going to preserve the cyclic ordering and it that So the action preserves Okay, and there's that there's going to be a fuchsian group. This is what footage proves Sort of uses tsukias. So remember they prove that I can think of it as acting on h2 Okay in the right way Okay So bodice is splitting here Is going to be Okay, so there is Another type of vertex. So this gives you the equivalence classes. So these are going to be the non These are going to be the brown vertices These are going to be brown vertices. There's another type of brown vertex And that's things that aren't like this Okay, so that's the rigid vertices. So these are vertices that don't admit any other kind of splitting so The first two the only thing Is that the valence of this has got to be so whenever the valence is three Whenever I have a cut pair that the valence is three It has to be the only choice is that the number in it is two I can't have an infinite equivalence class with valence equal to three Okay, this also has two things in an equivalence class, but it's just valence two So there's sort of two cases when it's valence two. They can either be infinite or not Okay, okay, so Bodich All white chalk This gamma one ended in hyperbolic To s one then um, there exists Canonical so elementary splitting v e non-elementary with the edges Okay, the stabilizers of the vertices and edges are quasi-convex These are the elementary the white are two ended And they're determined they're exactly their equivalence classes With the size of the number of the equivalence class equal to two Okay, so from the boundary Okay, and the non-elementary vertices either cyclically hanging fuchsian cyclically hanging fuchsian as I described there Or they're rigid And you can see this you can get these rigid vertices by Forming the graph as much as you can and then you'll be able to see these edges pointing out to something That's not there and that needs to be in there So I won't say too much about that, but these don't split anymore They don't split relative to the edge groups that are there So he looked at the boundary and tells a lot of information about the group structure Okay, instead of doing another example Uh, let me talk about relatively hyperbolic groups since we spent a fair amount of time describing the boundary of a um Of a hyperbolic space hyperbolic metric space We can talk about a different type of action on a hyperbolic metric space that gives us a relatively hyperbolic group and its boundary and then one Like current line of research is to take a lot of the stuff that's known about hyperbolic boundaries like this splitting and try to Generalize that to relatively hyperbolic groups and understand what goes on in the slightly more complicated set of relatively hyperbolic groups Okay, so a relatively hyperbolic group pair UP so I think first of all, I gave it least one definition of this is um a group Which acts Geometrically finitely and I'll define this Actually, I should think about a group pair on some proper hyperbolic metric space Okay, and I'll define what that means in a minute, but I the boundary so just to keep in mind the boundary of the group pair sometimes called the bodage boundary Is just equal to the boundary of that proper hyperbolic metric space x Okay, it turns out well, I have to finish giving you the definition first It's not true that all such spaces are quasi isometric However, it is true that all their boundaries are homeomorphic. So alert Not true I However, it's still true. So I can't use the proof before is true That the boundary of g with a specific set of peripheral subgroups is well defined and that's due to bodage I won't go into that proof here Okay, so let me say what uh Let me say what I'll say a little bit here and then I'll have to define this is going to take me more than one board Okay, so geometrically finite action on x x So g is going to act Properly discontinuously and by isometries, but not necessarily co-compactly on um x such that um One Every element of the boundary is either and I'll say what these are in a minute Is either a conical element point Or a bounded parabolic point. I'll say what both of those are quickly in a minute and the these p so um The elements of p are exactly the parabolic subgroups Maximum parabolic subgroups. Let me tell you what these things are So for so this is a generalization of a hyperbolic group because I could add co-compactly here Okay, and then it would actually follow that every element of the boundary is a conical limit point A hyperbolic group is a relatively hyperbolic group where the set of peripheral subgroups Parabolic subgroups is empty All right, so let me just say what those two Things are and then I can start doing examples So what is a conical element point and a bounded parabolic point? Okay, so a group p is parabolic. So um One thing That might confuse you if you're either used to it or you're reading a different source I write p for the whole collection. It's a collection of conjugacy classes finite number of conjugacy classes Sometimes people put one element of the conjugacy class. It's actually more so they get like p1 through pn This is I find this a little bit easier to deal with but the that's Just a different notation that she sometimes Okay, so parabolic It's infinite and contains no loxodromyx p a subgroup he's a loxodromyx And fixes some point On the boundary and it's bounded um If the boundary of x minus this point Modular the action of p is compact so for example If I have z goes to z plus one and upper half space Okay, if I think of this as h2 right So the boundary of h2 Equals s1 when I remove this fixed point the point of infinity is fixed by this Then what happens is the boundary minus a point Is just a line and when I roll it up Modular this group z that takes an infinity. I just get a circle Okay, so that's a bounded parabolic group the group generated by this Okay, and y contained in Um and so these parabolic point a bounded parabolic point is the xp So if this is a parabolic group bounded parabolic point is some such xp And then y contained in boundary x is a conical on that point if Of this action of g if there exists a sequence And two points a and b such that gi of y Is going to approach a So this is my point y so you could think of y being equal to a and that's a good example And gi leaving it fixed for example y equals a this point I have some Say loxadromic This b in the action kind of like scoots everything up like that Okay, uh g i of x equals b Sorry approaches b if um x is not equal to y in the boundary Okay, so this is uh Some of the conical limit points will be the end points of loxadromics And that's the first ones to think about if you haven't um seen that before So every point is like this and this is called a there's a definition of relatively hyperbolic So Let me give you some examples because I think examples are actually sometimes better than the definition Uh-huh. Can you really see that? How's that? I'll put my examples over here Down here, okay that I haven't used yet because I have I promise to use every word All right, let's do examples Okay I didn't have my examples now, but I want to do them now All right, so I've got One example and then I'll do the other one now. So here's just an example. So this is the boundary of g so g is a free group With no parabolic so I can think of that as that's just going to be the hyperbolic boundary That's going to be so this is a nice space. Here's my Action Here's some fundamental domain for my action My a is maybe going like this might be is going this way and I've got this boundary So the boundary is exactly the same as the hyperbolic boundary in this case Just equals a cantor set Okay, there's another action of the free group on h2 Okay, where I get the quotient is going to be a cusp hyperbolic group And here I'm going to have this These end points so these elements that these are lifts of This curve here. So these are now geodesics. They're going to be become parabolic so my Fundamental domain like look like this Okay, and that's actually going to act my space here It's going to be all of h2 so this is boundary of I'll call this group a b and now I'm going to switch notations and the peripheral subgroup is going to be the peripheral subgroups are going to be all the Conjugates of the commutator of a and b these are going to be the maximal parabolic subgroups. They'll fix some point There's a z so if I remove that point I'm going to get a line and the action of z is going to roll that up into a circle Okay, so this just equals s1. I'll do one more in a minute So this is a this is an example where the group itself is hyperbolic not always true that the group itself is hyperbolic um, let me do one more example and then I'll So one reason why I like hyperbolic groups is if you like hyperbolic not compliments It might disturb you to know that the fundamental group of some hyperbolic not compliments are not hyperbolic so if I look at s3 minus some not When it's a hyperbolic not there's many. Um, so that means that s3 minus this not admits uh geometrically finite action on H3 it's in particular its universal covers h3. Okay, but it just removes a knot at infinity. So Yes, uh s3 I'm going to say emits a hyperbolic structure with cusps So in particular the fundamental group acts on h3 that's a hyperbolic structure with cusps, so pi1 of s3 minus k modulo is is um acts by isometries on h3 and this action is a relatively hyperbolic action so um the boundary of pi1 of s3 minus k now I have to say what the peripheral subgroups these are going to be all the conjugates of these z plus z So a little neighborhood of this knot is going to be an essential torus That's going to be stabilized by z plus z. This is just equal to s2. Okay That's it, but let me tell you the relationship. So this is a non-hyperbolic group This group is hyperbolic, but I can still put different relatively hyperbolic structures on it and the boundary changes So let me just say what tran did um relating these or he probably wasn't the first one but he has a really nice exposition of this so when can I take a hyperbolic group and Make some of the subgroups parabolic and get a relatively hyperbolic group So we're gonna put that okay, so a collection of subgroups h1 to hn of a hyperbolic group is almost malnormal If uh, yeah, it can be in any group. I'm only thinking about hyperbolic groups right now If h i intersect g h j Uh g inverse so I take some con a conjugate for any g and g is finite unless The obvious thing happens is that these are the same group and this is one of those unless i equals j And g is contained in h i Okay, so I don't want what you don't want to happen with parabolic subgroups is you don't want them to intersect The the different conjugates to intersect in an infinite group Because you this one's going to fix this point and this other one's going to fix that point It's the maximal things that fix that point. So I can't have them intersecting Okay, so um boat it showed that if G is hyperbolic hyperbolic group and um p Is an almost normal collection Uh, thank you almost malnormal Of quasi convex subgroups. Now, you know what quasi convex is Consisting of finitely many conjugacy classes Then g p is relatively hyperbolic Okay, so for example in here, I've got this z. There's one conjugacy class and that's quasi convex It's malnormal. They the conjugates don't intersect at all and the The group when I use that as my peripheral subgroups is relatively hyperbolic. That's a relatively hyperbolic group pair It's a relatively hyperbolic group pair Okay, and then tran showed other people showed this too, but tran has a particularly nice exposition manning also has a Nice, so there's other people that if um g is hyperbolic And uh g p is also hyperbolic So is uh relatively hyperbolic Then the boundary the boatage boundary of that relatively hyperbolic pair Is equal to the boundary of g where i'm going to take um I'm going to take the boundary of p is going to be identified to some point For all of the pyro bog subgroups for all p contained in Okay, so I take the boundary of g. I take all the boundaries So those are quasi convex subgroups their boundaries are going to be in there And i'm going to take each one. I'm going to collapse them to a point Okay, that's exactly what's happening here All the boundaries of what i'm going to make the peripheral subgroups those conjugates of the commutator there are all going to be those uh the end points of those Geotesics up there, and i'm going to collapse those to a point to get the boundary So i'm going to get a canter set i'm collapse all those end points, and I get a circle So another example, let me do one more example And then i'll say a conduction and then i'll finish is that okay in here Okay, so another another thing you can do even with this free group because you can get something that doesn't even split anymore So another structure that I could put on this. I could take a punctured surface times i for this free group You know if I could think of it as a Punctured surface, and I'm just going to draw this like this is a little easier to draw like this So punctured surface times i Okay, and so not only am I going to have that point be um Peripheral, so i've got all the conjugates of the commutator I'm going to think all the conjugates of a i'm also going to make those peripherals So i'm sort of cussing this off And i'm going to make all the conjugates of b also peripheral Okay, so you can do this in a hyperbolic three manifold its boundary will be planar You have to climb in group and in this case the boundary of Sorry up two And my peripheral subgroups are going to be all the conjugates of a All the conjugates of b and all the conjugates i'll just call it c Here this boundary is going to be the uploading gasket So what'll happen is you're going to you can imagine where those a and b are on those circles You can put the a you can pinch the inside the b you can pinch the outside So what you'll actually get is you'll get I'll just draw it like this C is a commutator. Yeah, maybe I should write that This is actually just a little computation that this is what you get Because a lot is known about the uploading gasket This is in uh polluzi has since keep policy and Walsh You can get this nice beautiful hyperbolic structure here Okay, and this is a this is an example this sort of relates to what was going on with the Bodice composition is because if I take something like this Where these aren't the where these aren't peripheral where they're just boundaries and I glue other stuff to it I'll get a rigid piece a very simple rigid piece Um It doesn't split anymore. I can't there's no annulus I can put in here that is not going to cut one of these annuli that I have So that's a little vague Relating those two things, but this is in this paper. Okay, so one thing is that this is some Three manifold here. So this is a hyperbolic three manifold cusp three manifold And its boundary is going to embed in s2 because it acts on h3 and the boundary of H3 is s2 so boundary Um, which in this case is a relative like hyperbolic boundary boundary embeds in s2 So you can form a conjecture about what kind of things uh will happen when you have the Right kind of boundary. So let me just write a conjecture then I'll finish So conjecture And this is a generalization Of conjectures a famous conjecture of canon And also conjecture that should be easier of capovitch and cliner And this is theirs is in the um the hyperbolic case This isn't a relatively hyperbolic grace if a non elementary. So that means I don't want an elementary To have two points its boundary to have two points Now an elementary relatively hyperbolic hyperbolic group pair Gp has planar. So that means Bodage boundary so planar means embeds in s2 There's planar bodage boundary That does not have cut points So there are examples where they're cut points and this is not true. So this is necessary that does Not have cut points Then g itself Is virtually So that means there's a finite index subgroup of g And there's examples where you need that too isomorphic Uh to a climbing group. So that means Discrete subgroup of psl2 Uh See Okay, so cut point is a global cut point not a look not a local cup All right, I'll we'll end there