2/2 Boundaries of Hyperbolic and Relatively Hyperbolic Groups
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GeometrieHyperbolische GruppeGruppentheoriePhysikalische TheorieWalsh-FunktionensystemGruppe <Mathematik>FlächentheorieKonvexe HülleGruppentheorieHyperbolische GruppeKonvexe MengeRandwertQuotientÜberlagerung <Mathematik>Minkowski-MetrikAlgebraische StrukturGrundraumInverser LimesGeodätische LinieKurveDifferenteIsometrie <Mathematik>Zusammenhängender GraphUntergruppeGraphKreisflächeMannigfaltigkeitLimesmengeGüte der AnpassungHeegaard-ZerlegungVorlesung/Konferenz
04:54
GruppentheorieMinkowski-MetrikZyklische GruppeFlächentheorieKurveRechter WinkelPunktFreie GruppeHyperbolische GruppeGraphfärbungRandwertKomplex <Algebra>Geodätische LinieQuotientCantor-DiskontinuumInverser LimesAlgebraische StrukturÜberlagerung <Mathematik>KreisflächeMannigfaltigkeitKonvexe MengeGleitendes MittelDeskriptive StatistikUntergruppeGreen-FunktionHyperbolischer RaumMengenlehreGrundraumVorlesung/Konferenz
09:42
Freie GruppeFlächentheorieKnotenmengeGraphGruppentheorieStabilitätstheorie <Logik>Überlagerung <Mathematik>Heegaard-ZerlegungKubischer GraphUntergruppeInverser LimesKreisflächeKurveRandwertRechter WinkelRangstatistikMengenlehreGebäude <Mathematik>PunktrechnungLimesmengeGrundraumVorlesung/Konferenz
15:58
GruppentheorieÜberlagerung <Mathematik>GruppenoperationRandwertTopologieStellenringHeegaard-ZerlegungPunktRechter WinkelKnotenmengeSchnitt <Mathematik>GammafunktionGraphQuotientDifferenzkernEinsNumerische MathematikBipartiter GraphMengenlehreInverser LimesPunktrechnungEigentliche AbbildungHelmholtz-ZerlegungStabilitätstheorie <Logik>MultigraphUntergruppeArithmetisches MittelAlgebraische StrukturGeodätische LinieInvarianteFinitismusVorlesung/Konferenz
22:14
Topologischer RaumÄquivalenzklassePunktStellenringMengenlehreSchnitt <Mathematik>RandwertEinfach zusammenhängender RaumSortierte LogikKompakter RaumMinkowski-MetrikZusammenhängender GraphVorlesung/Konferenz
24:04
ÄquivalenzklasseAlgebraisch abgeschlossener KörperKlasse <Mathematik>Cantor-DiskontinuumPunktRechter WinkelLimesmengeZusammenhängender GraphRandwertFreie GruppeTeilbarkeitWechselsprungSortierte LogikKreisflächeSchnitt <Mathematik>EinsStabilitätstheorie <Logik>KnotenmengeÜbergangWärmeleitfähigkeitArithmetisches MittelEbener GraphMengenlehreGruppe <Mathematik>GruppentheorieInverser LimesUntergruppeHyperbolische GruppeFinitismusParabel <Mathematik>IndexberechnungDiskrete GruppePunktrechnungKappa-KoeffizientTopologieVorlesung/Konferenz
28:28
Lokales MinimumKappa-KoeffizientMetrischer RaumÄquivalenzklasseRandwertKlasse <Mathematik>MultiplikationsoperatorHyperbolische MannigfaltigkeitHyperbolische GruppeGruppenoperationGibbs-VerteilungDifferenteSondierungPunktrechnungSchnitt <Mathematik>PunktSortierte LogikHeegaard-ZerlegungAuswahlaxiomKnotenmengeGruppentheorieUnendlichkeitsinc-FunktionNumerische MathematikGraphBeweistheorieCantor-DiskontinuumStabilitätstheorie <Logik>Ordnung <Mathematik>Algebraisch abgeschlossener KörperStrömungsrichtungStellenringGeradeAbstimmung <Frequenz>Algebraische StrukturSpieltheorieBillard <Mathematik>Hyperbolischer RaumGraphfärbungMinkowski-MetrikVorlesung/Konferenz
37:06
Vorzeichen <Mathematik>UntergruppeRandwertBeweistheoriePunktUmwandlungsenthalpieInverser LimesMengenlehreGruppentheorieHyperbolische GruppeInnerer AutomorphismusHauptkrümmungKlasse <Mathematik>FinitismusElement <Gruppentheorie>ZahlensystemGruppenoperationGerichteter GraphRestklasseEigentliche AbbildungKegelschnittMinkowski-MetrikRechter WinkelEndlichkeitParabel <Mathematik>Güte der AnpassungOrdinalzahlHyperbolische MannigfaltigkeitGebundener ZustandUnendlichkeitIdealer PunktMetrischer RaumKreisflächeIsometrie <Mathematik>GeradeFolge <Mathematik>DifferenteLokales MinimumLoxodromeVorlesung/Konferenz
45:44
UntergruppeRestklasseRandwertGruppentheorieFundamentalgruppeTopologieAlgebraische StrukturInnerer AutomorphismusZahlensystemPunktParabel <Mathematik>GruppenoperationHyperbolische GruppeFundamentalbereichMinkowski-MetrikEinsInverser LimesFinitismusInverseGrundraumKegelschnittGeodätische LinieKreisflächeHyperbelUnendlichkeitLoxodromeGeradeUnendliche GruppeKnoten <Statik>Freie GruppeTorusQuotientKurveElement <Gruppentheorie>VertauschungsrelationKonvexe MengeLokales MinimumCantor-DiskontinuumNachbarschaft <Mathematik>MathematikIsometrie <Mathematik>Überlagerung <Mathematik>Vorlesung/Konferenz
54:21
GruppentheorieDiskrete GruppeArithmetisches MittelRandwertInnerer AutomorphismusHeegaard-ZerlegungHyperbolische GruppeIndexberechnungFinitismusSchnitt <Mathematik>Sortierte LogikKreisringEbener GraphPunktMultiplikationsoperatorSierpinski-DichtungBerechenbare FunktionUntergruppeAlgebraische StrukturFreie GruppeFlächentheorieStellenringRechter WinkelCantor-DiskontinuumMannigfaltigkeitGeodätische LinieVertauschungsrelationKreisflächeHelmholtz-ZerlegungVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:15
Okay, so I do want to clear up. A couple of people asked me, the convex hull in general is not convex. So just so you know. It's also called, some places in the literature,
00:28
the weak convex hull. So don't. Me too. I just want to clear that up. Okay, so let's do some examples. So I want to talk about Brodish's splitting.
00:40
It is quasi-convex though? Yes, it is quasi-convex. I might have said that, but I called it the convex hull. Okay, so I want to talk about Brodish's. So what is Brodish that you can see in the boundary? So remember, you can see the boundary of quasi-convex subgroups. And so Brodish is going to see a lot of quasi-convex subgroups
01:12
and how they are put together. So let me just do an example. Okay, so let's do an example.
01:28
Okay, so I'm going to start with a space and my group's going to be the, put them on a group of that space. So I'm going to take three surfaces with boundary. I'm
01:47
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, and I'm going to glue
02:01
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
02:26
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 them kind of going around swinging like this. It's a little easier to think of the
02:41
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,
03:03
and by Thurston, because this is Haakon, I don't need to use Perlman. This is realized by a geometrically finite, which is quasi-convex, climbing groups. I can realize this as
03:21
the take the convex hull 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
03:41
set, take the convex hull, 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, so it's hyperbolic, good example of a hyperbolic
04:00
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,
04:26
so this admits a graph of groups decomposition, where I have like a 4, f2, and f6 corresponding to these surfaces, and they're all glued together along the z. Okay, these z subgroups
04:44
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
05:08
on which this acts. One is to get a geometrically finite climbing 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
05:23
complex is going to have these little pieces of hyperbolic 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, so here, these are the lifts of this boundary curve,
05:51
so here's a structure for f2. 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
06:05
is going to be a Cantor set, which is going to be naturally embedded in this circle. All right, so this is the space, here's my space x, on which this hyperbolic group, one of
06:24
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 Bodish says that these are actually also quasi-convex subgroups, and those boundaries are going to appear, so they're going to be
06:43
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 I take some elevation of the curve a, it's left invariant by this cyclic group, I'm going to call g sub a, this is going to roll up to give me that
07:01
circle there, and these, I'm going to have these little Cantor 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,
07:27
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 Cantor set on the boundary.
07:42
Sorry, I didn't follow, so you take your, your kind of slash to go up, 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. Okay, like a book. Exactly, okay, so the endear's description is better than the drawing, so each of these
08:03
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 the 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
08:20
picture here, so let me draw the last one out here. I'm going to have a Cantor 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.
08:42
Okay, and then I'm not going to draw it, but out of 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 realized I have color.
09:03
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 on the third sheet, let's make this blue,
09:23
okay, so there's three sheets coming out, and then from this orange sheet, there'll be a green sheet and a 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,
09:59
all these are going to map down to say the orange surface, the blue surface,
10:09
and the red surface. Okay, question, George? Can you say you're going to glue more of that at some point? Okay, so I just drew one elevation of this surface, so every, every elevation of this
10:24
circle is going to be a geodesic, and each one is going to have three of these things going off from it. Each of these values are fine. Exactly, that's exactly right. Thank you, sorry, trying to keep, he's valiantly trying to keep me from killing myself.
10:42
Okay, so, so this is what this looks like. So now I can take this 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,
11:04
blue surface coming out, and I've got a green surface coming out, and the limit sets of these quasi-convex subgroups are going to be cantor sets,
11:22
which I'll see in the boundary, because these are all free groups, their boundaries are all free groups. Now I can form a graph by how these things are glued together. I'm just thinking in the universal cover now.
11:42
Bodish can think in the boundary, but let's just think in the universal cover. So I've got these elementary subgroups, and I've got these two-ended subgroups, and whenever I see one of those, I'm going to put a white, a white vertex. Okay, so the two-ended, stabilized by a two-ended group,
12:04
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. So I'm looking at this universal cover.
12:23
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.
12:42
Okay, then I have all the elevations of my surfaces, 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 in each one of those. Okay, and then I'm going to connect, I'm going to connect the white vert,
13:03
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 the stabilizer of this subgroup
13:22
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.
13:42
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, before I have these, I'll just draw all these brown.
14:01
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.
14:22
Okay, and they're all glued to this circle. Okay, so they're all going to be, like this white vertices,
14:49
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 bass there to the splitting, if you know what that is.
15:11
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.
15:23
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 4. Okay, and same.
15:40
So I'm going to have all these stabilizers going along, and then 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 showed. Okay, I think I've had that up for two minutes, but maybe I'll write it down over here.
16:05
So definition, elementary splitting. So you can think of 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,
16:22
give you the vertex groups. So definition, the elementary splitting is a minimal,
16:42
so this just means it doesn't fix a proper, there's no proper invariant subtree. Co-finite action on a simplicial bipartite tree.
17:15
Okay, and I want my tree is going to equal the,
17:24
I'm not going to call it a tree, because I'm going to use t maybe later, hopefully not. Okay, ve, vne, 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.
17:42
v sub e vertices are stabilized by, and I will sometimes confuse a vertex with its stabilizer, so I'll talk about that vertex,
18:02
and I mean if I'm talking group theoretically, that's the stabilizer, and the v and ne vertices have some other stabilizers, not two-ended.
18:24
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. You might be more used to. Okay, so these, the vertices, so bodich shows,
18:43
so what did bodich do? So bodich, given a group that was one-ended, so gamma one-ended, it's hyperbolic, and it's not, I don't want to deal with
19:04
virtually Fuxian 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,
19:23
look at the boundary to get elementary splitting. It'll actually be maximal and canonical, but let's just think how we get an elementary splitting.
19:43
Okay, and for this example right here, the splitting that he gets is exactly this one, this tree with this action on the tree. So let's talk about how he does that from the limit set or the boundary.
20:24
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 endpoints of that elevation of A, so that long white geodesic, when I remove that whole geodesic,
20:43
it's gonna split the universal cover into three pieces. Okay, so and if you look at those two points on the boundary, when I remove those two points on the boundary, it's gonna split the boundary into three pieces because I've got these three disjoint sheets going off to infinity.
21:02
Okay, so that's called a cut pair. And so if you look in this boundary, you can go around, you can see all these cut pairs and those are all gonna correspond to this elementary splittings. So let me just, I need to say a little bit of, let's look at the local cut points.
21:26
Okay, so the valence, so I'll just call that, okay, so the valence of X where X is a local, so what does it mean to be a local cut point? It means that when I remove it, so that means minus this point X has more than one end.
21:52
Okay, so the valence of a local cut point
22:01
equals just the number of ends of, okay, so I'm just gonna take all the set of local cut points, so Brodich puts the equivalence relation on all the set of local cut points.
22:26
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 gonna have these, the boundaries of my candor sets
22:41
are also gonna be local cut points. Okay, so I'm gonna say that X is equivalent to Y if the valence of X is equal to the valence of Y and X, Y is a cut pair.
23:08
Okay, so for example, the two points at the end points of that geodesic upstairs are gonna have this. So each has valence three, and when I remove both of them, I'm gonna get three pieces, yeah? When you say more than one end, you mean connected components?
23:26
Yes, you take a compact exhaustion and you wanna say there's always gonna be so many, 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,
23:40
it's actually not gonna separate the boundary into other components. But that space is gonna have three ends. If I start looking at bigger and bigger compact sets, I'm gonna end up with like three ways to go to infinity, basically. So when I just remove one, it's not a cut point,
24:01
it's a local cut point. So if I just remove one, it's not gonna separate the boundary. But if I 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, then there's more than one component. Okay, so he looked at these equivalence classes.
24:21
So the equivalence classes. So when the size is two, they're gonna correspond to, they're gonna be stabilized by, so if the size of an equivalence class is equal to two,
24:42
the stabilizer of that equivalence class is z. And these correspond to the elementary vertices. So that's sort of the first thing to see.
25:12
And I can also have valence two, so that the cantor sets valence two
25:20
with infinitely many equivalent points. And the closure of such an equivalence class,
25:40
closure of equivalence class is gonna 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 elevation of G, you can clearly see those three sheets coming out.
26:02
But if I remove the cantor set, what's gonna happen? Let me draw a little magnification of what's going on at the cantor set level. If I have, say an orange cantor set,
26:24
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. And I'll actually have a whole green one
26:42
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.
27:02
Okay, this one. I'm gonna write that out more formally in a minute. I just wanna give you an idea. Okay, so what's gonna happen is like,
27:20
two points like this on these cantor sets that aren't jump points, so the jump points are what you can kind of see, it's gonna 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 gonna divide this boundary into the piece in between here on this.
27:42
This is sort of looks from far away like a circle and that's actually kind of accurate here. Divide that into this piece right here and then this other whole other mess. Okay, so these are gonna be all these, the non-jump points of the cantor set are going to be these, are gonna be valence two and the closure of that class
28:01
is gonna be this whole orange cantor set and that is gonna 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
28:22
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 Bowdich said here. I'm gonna use this board since I'm trying to use all my boards.
28:44
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. Bowdich's paper on splittings and cut points
29:02
is very good. There's also 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 you if you want. It has exactly this example
29:22
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 gonna be equal to two.
29:40
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 its valence is three, and that's a cut pair, then it's gonna stabilize a two-ended group. So these are stabilized by two-ended groups.
30:04
So notice I'm kind of going back and forth between the cut pair and the vertex because a cut pair is gonna correspond to a vertex,
30:22
one of those white vertices up there. So those are gonna be like white vertices. So correspond to white vertices. Okay, I could have the valence of x could be two and the number of x just equals two still.
30:43
Okay, so that's also gonna be a white vertex. And then I could have the valence, valence of x be equal to two and the number in the equivalence class be infinite.
31:00
Okay, when I have such an equivalence class like this, this is gonna correspond to a cyclically hanging Fuxian group. So there's a, so I take the closure of that equivalence class, I should do it like that. And this is gonna map into, so this is gonna be a Cantor set
31:21
and it's gonna map into a circle and that the stabilizer is gonna preserve the cyclic ordering.
31:50
Okay, so the action on this equivalence class is going to preserve the cyclic ordering and that, so the action preserves.
32:13
Okay, and there's going to be a Fuxian group. This is what Boterich proves. Sort of uses two kias. So remember they proved that.
32:25
I can think of it as acting on H2. Okay, in the right way. Okay, so Boterich's splitting here is gonna be,
32:40
okay, so there is another type of vertex. So this gives you the equivalence classes. So these are gonna be the non, these are gonna be the brown vertices. These are gonna be brown vertices. There's another type of brown vertex and that's things that aren't like this.
33:01
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 gotta be, so whenever the valence is three, whenever I have a cut pair that the valence is three,
33:21
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 Boterich,
33:50
I'll white chalk, is gamma one-ended and hyperbolic to S1.
34:11
Then there exists a canonical elementary splitting,
34:28
VE, non-elementary with the edges. Okay, the stabilizers of the vertices and edges are quasi-convex, okay.
34:53
These are the elementary, the white, R2-ended,
35:08
and they're determined, they're exactly their equivalence classes with the size of the number
35:21
of the equivalence class equal to two. Okay, so from the boundary, okay, and the non-elementary vertices are either cyclically hanging function,
35:45
as I described there, or they're rigid. And you can see this, you can get these rigid vertices
36:00
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.
36:23
So he looks at the boundary and tells a lot of information about the group structure. Okay, instead of doing another example, let me talk about relatively hyperbolic groups. Since we spent a fair amount of time describing the boundary of a hyperbolic space,
36:43
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 current line of research is to take a lot of the stuff that's known about hyperbolic boundaries,
37:01
like the 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,
37:33
I think Francois gave at least one definition of this, is a group which acts geometrically finitely,
37:44
and I'll define this. Actually, I should think about a group pair on some proper hyperbolic metric space.
38:18
Okay, and I'll define what that means in a minute. But the boundary, so just to keep in mind,
38:22
the boundary of the group pair, sometimes called the Bodege 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
38:42
are homeomorphic. So alert, not true, RQI. However, it's still true,
39:01
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.
39:23
And that's due to Bodege. I won't go into that proof here. Okay, so 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.
39:41
Okay, so geometrically finite action on X. So G is going to act properly discontinuously
40:09
and by isometries, but not necessarily co-compactly on X. Such that, one,
40:26
every element of the boundary is either, and I'll say what these are in a minute, is either a conical element point
40:46
or a bounded parabolic point. I'll say what both of those are quickly in a minute.
41:01
And these P, the elements of P are exactly
41:21
the parabolic subgroups, maximal parabolic subgroups. So let me tell you what these things are. So this is a generalization of a hyperbolic group,
41:42
because I could add co-compactly here. Okay, and then it would actually follow that every element of the boundary is a conical element point. Hyperbolic group is a relatively hyperbolic group
42:00
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?
42:33
Okay, so a group P is parabolic. So one thing that might confuse you if you're either used to it or you're reading a different source,
42:41
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. I find this a little bit easier to deal with, but it's just a different notation that's used sometimes.
43:04
Okay, so parabolic, it's infinite and contains no loxodromics, P a subgroup.
43:24
Here's no loxodromics and fixes some point. And the boundary, and it's bounded
43:40
if the boundary of X minus this point, modulo the action of P is compact. So for example, if I have Z goes to Z plus one in upper half space, okay, if I think of this as H2, right, so the boundary of H2 equals S1.
44:05
When I remove this fixed point, the point at infinity is fixed by this, then what happens is the boundary minus a point is just a line. And when I roll it up, modulo this group C
44:27
that fixes infinity, I just get a circle. Okay, so that's a bounded parabolic group, the group generated by this. Okay, and Y contained in, and so these a bounded parabolic point is the XP.
44:44
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
45:03
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.
45:24
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 at fixed, for example, Y equals A. At this point, I have some, say, loxidromic,
45:41
this B in the action kind of scoots everything up like that. Okay, GI of X approaches B if X is not equal to Y in the boundary. Okay, so this is,
46:05
some of the conical limit points will be the endpoints of loxidromics, and that's the first ones to think about if you haven't seen that before. So every point is like this, and this is called a, there's a definition of relatively hyperbolic.
46:20
So let me give you some examples because I think examples are actually sometimes better than the definition. Can you really see that? How's that? I'll put my examples over here down here, okay,
46:41
that I haven't used yet, because I promised to use every word. All right, let's do examples. I didn't have my examples now, but I want to do them now. All right, so I've got one example,
47:07
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 parabolics. So I can think of that as, that's just going to be the hyperbolic boundary.
47:22
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, my B is going this way,
47:41
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,
48:03
and here I'm going to have these end points. So these elements, that these are lifts of this curve here, so these are now geodesics. They're going to become parabolic.
48:22
So my fundamental domain look like this, and that's actually going to act. My space here is going to be all of H2.
48:41
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.
49:02
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 an example where the group itself is hyperbolic, not always true that the group itself is hyperbolic.
49:20
Let me do one more example, and then I'll, so one reason why I like hyperbolic groups is if you like hyperbolic not complements, it might disturb you to know that the fundamental group of hyperbolic not complements are not hyperbolic. So if I look at S3 minus some not,
49:49
minus a hyperbolic not, there's many. So that means that S3 minus this not admits
50:02
a geometrically finite action on H3. In particular, it's universal covers H3. Okay, if I just remove the not at infinity. So yes, S3, I'm going to say,
50:20
admits a hyperbolic structure with cusps. So in particular, the fundamental group acts on H3. That's a hyperbolic structure with cusps. So pi one of S3 minus K modulo is
50:49
acts by isometries on H3. And this action is a relatively hyperbolic action.
51:01
So the boundary of pi one of S3 minus K, now I have to say with the peripheral subgroups, these are going to be all the conjugates of these z plus z's. So a little neighborhood of this not is going to be an essential torus that's going to be stabilized by z plus z. This is just equal to S2.
51:20
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 relating these or he probably wasn't the first one, but he has a really nice exposition of this.
51:50
So when can I take a hyperbolic group and make some of the subgroups parabolic and get a relatively hyperbolic group? So a collection of subgroups H1 to HN
52:17
of a hyperbolic group is almost malnormal.
52:36
It can be in any group.
52:41
I'm only thinking about hyperbolic groups right now. If H I intersect G H J G inverse, so I take some conjugate for any G in G is finite unless the obvious thing happens
53:07
is that these are the same group and this is one of those. Unless I equals J and G is contained in H I.
53:21
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 different conjugates to intersect in an infinite group because 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 Bowditch showed that if G is hyperbolic,
53:50
hyperbolic group and P is an almost normal collection.
54:11
Thank you. Almost malnormal of quasi-convex subgroups. Now you know what quasi-convex is
54:28
consisting of finitely many conjugacy classes
54:51
and G P is relatively hyperbolic.
55:02
Okay, so for example in here I've got this Z. There's one conjugacy class and that's quasi-convex. It's malnormal. The conjugates don't intersect at all and the group when I use that as my peripheral subgroups is relatively hyperbolic. That's a relatively hyperbolic group pair.
55:22
It's 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 G is hyperbolic and G P is also hyperbolic,
55:47
so is relatively hyperbolic, then the boundary, the bodege boundary,
56:01
of that relatively hyperbolic pair is equal to the boundary of G where I'm going to take the boundary of P is going to be identified to some point for all of the parabolic subgroups, for all P contained in.
56:24
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 and 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 as conjugates
56:40
of the commutator there are all going to be the endpoints of those geodesics up there, and I'm going to collapse those to a point to get the boundary. So I'm going to get a Cantor set. I'm going to collapse all those endpoints and I get a circle. So another example, let me do one more example, and then I'll say a conjecture and then I'll finish. Is that okay in here?
57:03
Okay, so another thing you can do even with this free group is 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.
57:20
Even if I could think of it as a punctured surface and I'm just going to draw this like this. It's a little easier to draw like this. So punctured surface times I.
57:41
Okay, and so not only am I going to have that point be 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 peripheral, so I'm sort of cussing this off and I'm going to make all the conjugates of B also peripheral.
58:02
Okay, so you can do this in a hyperbolic three manifold. Its boundary will be planar. You have to incline in group. And in this case, the boundary of, sorry, of two and my peripheral subgroups are going to be all the conjugates of A, all the conjugates of B and all the conjugates,
58:21
I'll just call it C here. This boundary is going to be the uploading gasket. So what will 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.
58:45
C is a commutator. Anyway, 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 Poluszky, Poluszky and Walsh.
59:05
You can get this nice, beautiful hyperbolic structure here. Okay, and this is an example. This sort of relates to what was going on with the Bode she composition is because if I take something like this where these aren't peripheral, where they're just boundaries
59:21
and I glue other stuff to it, I'll get a rigid piece, a very simple rigid piece. 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.
59:40
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 it's...
01:00:00
Boundary is going to embed in S2 because it acts on H3, and the boundary of H3 is S2. So boundary, which in this case is a relatively hyperbolic boundary. Boundary embeds in S2. So you can form a conjecture about what kind of things
01:00:22
will happen when you have the right kind of boundary. So let me just write a conjecture and then I'll finish. So conjecture.
01:00:42
And this is a generalization of conjectures, a famous conjecture of canon, and also a conjecture that should be easier of Kapovich and Kleiner.
01:01:04
And theirs is in the hyperbolic case. This is in a relatively hyperbolic case. If a non-elementary, so that means I don't want an elementary to have two points, this boundary would have two points.
01:01:20
Now an elementary, relatively hyperbolic group pair, GP, has planar, so that means Brodich boundary,
01:01:46
so planar means embeds in S2, has planar Brodich boundary that does not have cut points.
01:02:03
So there are examples where there are cut points, and this is not true, so this is necessary, that does not have cut points. Then G itself is virtually,
01:02:24
so that means there's a finite index subgroup of G, and there's examples where you need that too, isomorphic to a Kleining group. So that means a discrete subgroup of PSL2. Let's see.
01:02:44
Okay, so a cut point is a global cut point, not a local cut point. All right, we'll end there.