1/2 Boundaries of Hyperbolic and Relatively Hyperbolic Groups
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00:00
Formation <Mathematik>MengenlehreHyperbolische GruppeObjekt <Kategorie>UnrundheitGeodätische LinieMetrisches SystemMetrischer RaumRandwertTopologieMultiplikationsoperatorGibbs-VerteilungHyperbolischer RaumQuantenzustandPunktHyperbolische MannigfaltigkeitParametersystemVorlesung/Konferenz
02:25
GammafunktionPunktHyperbolischer RaumMinkowski-MetrikAbstandQuantenzustandInverser LimesBeweistheorieHausdorff-MetrikDifferenteNachbarschaft <Mathematik>TopologieUnendlichkeitMetrischer RaumRestklasseÄquivalenzklasseGeodätische LinieOrbit <Mathematik>Produkt <Mathematik>GraphGruppenoperationZeitrichtungFolge <Mathematik>MereologieMultiplikationsoperatorNumerische MathematikSortierte LogikEinflussgrößeMetrisches SystemGebundener ZustandKlasse <Mathematik>RandwertAbgeschlossene MengeStichprobenfehlerVorlesung/Konferenz
09:22
RechteckRandwertÄquivalenzklasseAbstandPunktIsochoreMinkowski-MetrikHyperbolischer RaumMetrischer RaumUnendlichkeitKlasse <Mathematik>BimodulGraphFolge <Mathematik>TopologieInverser LimesGruppenoperationMomentenproblemZahlensystemIsometrie <Mathematik>Hyperbolische MannigfaltigkeitDifferenzkernProdukt <Mathematik>Geodätische LinieLokales MinimumHomomorphismusSpezielle unitäre GruppeMengenlehreHyperbolische GruppeHomöomorphismusSupremum <Mathematik>RestklasseVorlesung/Konferenz
16:18
Formation <Mathematik>RandwertIsometrie <Mathematik>Rechter WinkelMinkowski-MetrikÄquivalenzklasseMetrischer RaumRichtungGammafunktionPunktIsochoreProdukt <Mathematik>Klasse <Mathematik>KompaktifizierungStetige FunktionNachbarschaft <Mathematik>Weg <Topologie>RadiusAbstandBasis <Mathematik>TopologieGebundener ZustandGrenzwertberechnungGeodätische LinieFolge <Mathematik>GruppenoperationInverseDifferenzkernBijektionSupremum <Mathematik>QuantenzustandHomöomorphismusInverser LimesSummierbarkeitVorlesung/Konferenz
23:14
Nachbarschaft <Mathematik>Klasse <Mathematik>AbstandPunktPrimidealSortierte LogikProdukt <Mathematik>EinsRandwertMultiplikationsoperatorRechter WinkelGeodätische LinieFolge <Mathematik>Offene MengeUngleichungGeradeFormation <Mathematik>Gebundener ZustandGrenzwertberechnungTopologiePhysikalischer EffektÄquivalenzklasseVorlesung/Konferenz
29:50
Produkt <Mathematik>Numerische MathematikUngleichungMultiplikationsoperatorGeodätische LinieNachbarschaft <Mathematik>GeradeAbstandVorlesung/Konferenz
31:53
Algebraische StrukturUntergruppeHyperbolische MannigfaltigkeitRandwertGruppenoperationMinkowski-MetrikAbstandNachbarschaft <Mathematik>HomöomorphismusParkettierungGeodätische LinieFlächentheorieFolge <Mathematik>UnendlichkeitTeilmengeGraphHyperbelverfahrenGeometrieCantor-DiskontinuumInvarianteMatrizenrechnungEigentliche AbbildungGebundener ZustandBeweistheoriePunktAnalytische FortsetzungKonforme AbbildungUnendliche GruppeGüte der AnpassungTopologieHyperbolischer RaumMengenlehreFigurierte ZahlSymmetrieQuaderKontinuumshypotheseAusgleichsrechnungFreie GruppeIsometrie <Mathematik>IsochoreDifferenzkernMetrischer RaumProdukt <Mathematik>Hyperbolische GruppeVorlesung/Konferenz
40:50
UntergruppeMinkowski-MetrikLimesmengeMultiplikationsoperatorPunktHyperbolischer RaumGruppenoperationGeradeRandwertGeneigte EbeneGeschlossene MannigfaltigkeitOrbit <Mathematik>Inverser LimesKonvexe HülleGraphTeilmengeUnendlichkeitFolge <Mathematik>DurchmesserObjekt <Kategorie>GeometrieMengenlehreEndlich erzeugte GruppeHyperbolische GruppeÄquivalenzklasseKlasse <Mathematik>Konvexe MengeAussage <Mathematik>Abgeschlossene MengeEigentliche AbbildungFlächentheorieKreisflächeGeodätische LinieVorlesung/Konferenz
49:42
LimesmengeRandwertGruppenoperationUntergruppeFormation <Mathematik>Vorlesung/Konferenz
Transkript: English(automatisch erzeugt)
00:16
I'm gonna overlap a little bit with what Francois said, so just to review and get us all on the same page.
00:22
So I apologize if you know that well, but hopefully you'll enjoy that. Okay, so the boundary of a hyperbolic group. Okay, so in general, this is a topological object
00:40
that's gonna be canonically associated with a hyperbolic group, and it's gonna tell us a lot of information. So I'm gonna show that this is a canonical object or topological object. It actually also has a metric structure, which I won't go into today. And then I'm gonna show some things that it's good for. What does it tell you?
01:03
Tells you a lot. So the spoiler is a lot. Okay, and then I'm gonna talk a little bit about relatively hyperbolic parameters.
01:27
Okay, so this is gonna be a lot. So I wanna just quickly review this, and then I wanna spend most of my time on what the boundary of a hyperbolic group actually tells you about that hyperbolic group. Okay, so X is gonna be a proper,
01:44
proper is important. Geodesic isn't so important, but I'll usually be implicitly assuming it, hyperbolic metrics ways. So let me just give three very common definitions.
02:03
So definition one, boundary X is the set of geodesic rays from some point.
02:25
The topology won't depend on the point. And modulo equivalence. Okay, so what's that equivalence gonna be? So we want the geodesic rays to be equivalent if they're in the same house or
02:41
they have bounded house or distance. So let me just, a good example to keep in mind that I think you've done is H2, delta hyperbolic space. I'm gonna have rays that go out here. And if I think about it, maybe I have some slightly wiggly hyperbolic space and there's some other geodesic ray that stays the same. And I kinda think it ends at the same point.
03:01
If they stayed in the same house or distance, they would end at the same point in H2. Okay, so this is bounded house or distance. So I'll say that gamma one is equivalent to gamma two
03:20
if the image of gamma one is contained in the neighborhood of C of the image of gamma two and vice versa. There's some C that just works.
03:41
So this is a good definition just for getting your intuition. Something, so lots of times we're gonna be dealing with quasi-isometry, so we're not gonna wanna have geodesic rays. So definition two is really similar to this. This is one that I'll use. And that's, I'm thinking about quasi-geodesic rays.
04:00
So just quasi-geodesic rays at some x not contained in x, okay? And again, modular, the same equivalence.
04:25
And this is bounded house or distance. So you can just think about wiggly rays
04:41
and you want them to stay in some distance. And remember that in a delta hyperbolic space, if I have a quasi-geodesic, it's within bounded distance of some geodesic. Okay, so you can take a geodesic in this class if you have a geodesic metric space and just think about that. Okay, so let's think about definition three, which is the one I wanna talk a little bit about.
05:03
So now instead of thinking about the actual ray, I'm just gonna think about points going off to infinity. So this turns out to be really useful. Often we think of a group as just the orbit of its point acting on some delta hyperbolic space. So we might just wanna think about a bunch of points. Okay, it actually is very helpful. It's a little bit less intuitive, but it's very helpful.
05:25
And this is sequences tending to infinity. Okay, so I'm gonna look at,
05:40
so let me just say what that means, to tend to infinity. I'm gonna look at, so the Gromov product, which I think you've seen, at least Indira was probably telling you about it. So I have x, y at some point p. It's just defined to be one half the distance, I'm in a metric space,
06:01
x, p plus the distance y, p minus the distance x, y. So let's draw a little picture of this. So if I'm in hyperbolic space, I've got some p. I'm gonna look at my two points out here, and I'm gonna have the distance from x to p,
06:23
distance of y to p minus the distance between them. Okay, so I'm drawing a little picture in x, p. So notice when they get way far away, this distance is actually gonna be going to infinity. Okay, so these will be big. And if they're close together,
06:40
if they're sort of going to the same point, so this is sort of a idea of a proof that what happens if these are points along rays that are staying within bounded Hausdorff of each other, way out here, they're gonna look exactly the same. Okay, and this is gonna go to infinity. Okay, so and another way to think about,
07:01
a good way to think about the Gromov product, I think is in a tree. So if I have like, here's my p, and this is x. So I'm drawing part of the Cayley graph for F2, x and y.
07:25
Okay, so let's look at what the Gromov product is here. Okay, the distance from p to x. So like, this is just gonna be three, right? This is gonna be one, two,
07:42
one, I wanted this to be four, one, two, three, one, two, three. I'm gonna put another one here. Plus, okay, plus four minus five, right? So this is gonna be what?
08:02
One. Okay, so what is this? So where's my geodesic between x and y? So this is, it's fun to do this in a tree if you haven't done this for a while. So this distance in this space is gonna measure the distance from p to the geodesic from x to y, right? So here's my geodesic between x and y. I gotta go through that point, okay?
08:26
And then here's this distance to one, to that geodesic. Okay, so in a tree you can start to see, if I have, I wanna look at sequences going to infinity, so let me say what that means.
08:43
I'm gonna say that ai tends to infinity, or sometimes I'll actually write error infinity, but I'll have lots of different points on the boundary. If the limit as i goes to infinity
09:03
and j goes to infinity of ai, aj equals infinity. So that means whenever i and j are bigger than some number, then this is as big as I want it. Okay, I can make that as big as I want it.
09:20
So it's a good idea to play with that in a tree. Tree is zero hyperbolic, everything's kinda nice. Erasers, tricky.
09:43
Oh yeah, there's more boards here. Oh, there's even three. All right, I'll put some stuff on the side. Okay, so, and then in the exercises,
10:01
this shows that there's, you can think of this as like thin rectangles to be thin. So definition, maybe I'll put this over here. X is delta hyperbolic, if and only if,
10:21
for all x, y, z, w, x, y of w is greater than or equal to the minimum of x, z, w, y, z, w minus delta.
10:43
Okay, so you can think of, you can do everything. You could never know about geodesics and be happy in a hyperbolic metric space. Okay, i.e. thin rectangles. See the exercises.
11:12
Okay, and also in a delta hyperbolic space, let me just write this down. I have n x delta hyperbolic, y of p
11:25
is gonna be less than or equal to the distance between p and this notation here means the geodesic between x and y. And this is x, y plus delta. Okay, so in a zero delta hyperbolic space
11:43
is what you're starting to get thinking about the graph. Okay, so I'll leave these up here. I'm gonna use this in a minute. All right. Yeah?
12:03
Oh, I haven't said, I haven't finished. Thank you. So, very good. Modulo equivalence, let me write that down. So, I'll just write definition three and I'll say sequences tending to infinity. Now I've said what that means.
12:21
Now let me say modulo the equivalence to infinity
12:41
and then I want to say that these sequences are equivalent if the limit as i goes to infinity and j goes to infinity of ai bj equals infinity.
13:01
Okay, so this is an equivalence relation for a hyperbolic space. It's not necessarily equivalence relation if you don't have a hyperbolic space. Okay, so this, it's not completely obvious that this is transitive. Okay, so you need delta hyperbolicity.
13:22
Okay, and then this is my set. Thank you. Okay, so now what I want to do, so this is just the boundary of a hyperbolic metric space. I want to think about the boundary of a group. So the boundary of a group that acts geometrically on a delta hyperbolic metric space. That's the definition of a hyperbolic group,
13:42
but maybe it acts on different hyperbolic metric spaces. They'll all be quasi-isometric. I want the boundary to be well-defined. Not do I want just the set to be well-defined. I want the topology. So let me put a topology on it and then I want to show that this is, or give you some idea why this is well-defined for the quasi-isometry type of the space. So now let's put a topology
14:08
and as Anna mentioned yesterday, the, just like in H2, the isometries of the hyperbolic space are going to induce homeomorphisms of the boundary. And so that helps you see the action of the group
14:20
really helpfully. Okay, so what I want to do is I'm going to extend this grandma product to the boundary. Okay, so I have M and W. So M and N are points in the boundary.
14:51
Okay, this is going to be equal to the supremum over Xi tending to M. Okay, so what is Xi tending to M? That means the sequence Xi, it's equivalence class
15:02
under this equivalence as a point in the boundary is M. I won't write that, but that's what I mean by Xi tending to M. It means the equivalence class of the sequence Xi, I'm calling that M. Okay, because that's what points are on the boundary. And Yi tending to N of the limb M, Xi, Y, J
15:32
and W. Okay, so what that's supposed to be indicating is sort of like roughly the distance
15:41
to this kind of geodesic out here as I go out to it. That's a good way to think of this. Okay, why did I put supremum and M? Just to be a pain in the butt. There's actually examples that maybe I should have put in the exercises. But if you want, you can think about this space, which is quasi-isometric to Z.
16:01
So it has two points at the boundary. And you can find sequences going to positive and negative infinity so that this wouldn't be well defined unless you did this. Okay, so that's a fun thing to play with too. Okay, so you need that supremum and M. I've actually seen in the literature M of M. But you could have like zero and one going in that example.
16:22
Okay, so now let's put a topology. Let's put a topology on this space using the group union the boundary. Can you see this board under here?
16:48
Which equals X union boundary X. Okay, so I want a compactification. So if you think, if you're H3 person like me, you can think of X using the boundary as a solid ball. Okay, so I've got, if I have X is some point
17:06
in the big S, I'm gonna just look at epsilon balls. So I'm gonna look at a ball of radius R around X. So these are metric balls. This will be my, I'm doing a basis for the topology. Okay, and what about X contained in the boundary X?
17:23
We have a neighborhood of R of X equals Y such that XYW is greater than R. Okay, so some of these points Y. Okay, so what does this mean? The sum of these points Y could be in the boundary.
17:40
Right, okay, so I'm gonna have, I've got, eventually I'll draw an example that's not H2. So I have some point X in the boundary. Okay, so some of the points will be Y in the boundary such that this Gromov product from some distance is very far, is very large. Okay, and some of these also,
18:00
I wanna get a little neighborhood here. I wanna glue this boundary to the space. Okay, so some of these will be points. So if Y is contained in the boundary of X, it's the definition above, this definition of the Gromov product. Okay, above.
18:20
Okay, if Y is actually in X, then I use XYW. It's just gonna equal the supremum as XI goes to X of the limit of XI and YW.
18:55
Okay, so I just modified this definition so if one of them is in the space and one of them is at the boundary,
19:00
I don't need to take both sequences. Okay, because Y is actually in the space. Okay, all right, so this is my topology and my spaces gives me a neighborhood basis. Okay, so here's some facts.
19:28
I'm not gonna prove everything. Try to give you some idea why if I have a quasi-isometry of the spaces, we have a homeomorphism. Okay, so it turns out that when X is proper,
19:40
I did something wrong? So if X is proper, and I'm always assuming hyperbolic metric space, then we're gonna have that X hat and boundary X are compact. Okay, they're also house store. So DX is also house store.
20:06
This is not so hard to prove. That is also not so bad. So I'm gonna show that the map is continuous, that I have a continuous bijection when I have an isometry. Or show that, oh, now I need to use my hook. So we're gonna show that if our X
20:24
is quasi-isometric to X, there's a continuous map and it's bijection.
20:48
Okay, so let me just outline that using grandma product. Okay, so let's let,
21:07
so I'm gonna kind of go between definitions and so bear with me. So for the, just to define the map, I'm gonna use the definition of quasi-geodesics. So we're gonna let F takes X to X prime,
21:22
B quasi-isometry, B inverse G. Okay, so M, equivalence class in the boundary, is represented by quasi-geodesic ray, gamma.
21:53
Okay, so F, because this is a quasi-isometry, F of gamma is also quasi-geodesic ray.
22:02
Okay, so I'm gonna set boundary F of the equivalence class of gamma, just equal to the equivalence class of F gamma. Okay, so this will be in boundary X.
22:21
Okay, so now if, so I wanna show that, so if I have two things that are equivalent, right? So if they're in bounded distance, let me just say this real quick, I could just put this exercise. If the house-torped distance between these two is less than K, okay, then the house-torped distance between gamma one, F of gamma two,
22:45
is gonna be less than or equal to, say, lambda K plus C. Okay, so then this is a well-defined map. Okay, so if I look at the, and the inverse is also bounded,
23:00
G composed with F of gamma and gamma, okay, is bounded. Okay, so this is gonna give me a bijection. So, oh, now I get to use my hook.
23:28
That's pretty good. That's actually way better than the ones that have buttons, because buttons, you can get mixed up.
23:41
Now I'm just gonna be messing with these all the time. All right, so you know the definition now. Oh, you can't, which one do you want? All right. Oh, you know what I should do?
24:01
I should do this.
24:23
Okay, so now let's look at, I wanna show that the pre-image of OpenSet is open. I'm gonna go back to my Gromov product definition, where I used its topology.
24:42
All right, so let's look at some Y contained in. I wanna show that if I have this, like, you can sort of think of this as like a delta, but that goes the other way. I want it to be big. Is gonna imply that boundary F,
25:04
this is my map of Y, is contained in some other little neighborhood of boundary F of X. Okay, so I wanna be able to get my neighborhood small enough around here, so that I can get inside some open neighborhood in here.
25:22
Okay, just like delta epsilon definition. So let's look at the Gromov product. So I have F of xi, F of yi. So here I'm thinking xi is gonna be tending to x, this is equivalence class, and yi is gonna be tending to y. Doesn't matter which sequences I choose.
25:45
Okay, so I'm gonna see how, what this neighborhood looks like. I'm gonna mess around with it to get something, thinking about how close I need to get the sequences x and y to be. Okay, this is F of x naught, greater than or equal to the distance F of x naught.
26:04
This means the geodesic F of xi, F of yi minus delta. Okay, so that comes from this fact that I, this inequality right here.
26:23
Okay, so this means the geodesic between F of xi and F of yi, okay? And this just equals, so there's some point on this geodesic that realizes this. So this is just equal to the distance between F of x naught and t prime
26:43
minus delta for some t prime on, okay, cause it's just realized.
27:01
Okay, so the image of this, of xi, yi, is gonna just be a quasi-geodesic. So this is the geodesic, but there's some quasi-geodesics, that's the image of the geodesic between xi, yi. So this is going to be greater than or equal to, which is within bounded distance of this geodesic.
27:21
Okay, this is gonna be the distance between F of x naught, F of t. Okay, so this is some point on the geodesic minus k minus delta, okay? Where the actual geodesic between F of xi and F of yi
27:43
is in the bounded distance of the image of the geodesic between xi and yi, let me write that out. So for some t, because this maps to a geodesic.
28:03
This maps to a quasi-geodesic. Uh-huh? No, this geodesic is in sum, this is sum, but this is not any particular k,
28:20
this is just, it's within some bound. So it's just some k. So it's not the k, if I used that k before, it's just, it's in a bounded distance. Okay, and this k depends on that, what that quasi-geodesic constant is. I'm just not worrying about it right now. Okay, so this is gonna be greater than or equal to, now I'm gonna use it,
28:40
this is my quasi-geodesic, distance between x naught and t minus c. So there's a quasi-geodesic, okay? Minus k minus delta, okay? So I just carry those down.
29:02
And this is gonna be equal to one over lambda, okay? The distance between x naught and x i, y i, because this t was in x i, y i,
29:26
minus c, minus k, minus delta. Okay, now, okay, so what did that tell us?
29:40
So look, we related the Gromov product of the image to this Gromov product of the original thing. All right, maybe I missed a line. Oh yeah, I did miss a line, let me put it on this thing. Okay, so remember this,
30:00
the distance to the geodesic between x i and y i is bounded by the Gromov product. So this is gonna be, this is x naught. This is again from that inequality over there.
30:22
Minus c plus k plus delta. Okay, so this relates to the Gromov product of the image to the Gromov product of the original thing. So we have, if that Gromov product
30:41
is bigger than that number, x naught is gonna be greater than lambda times n, plus c, plus k, plus delta.
31:01
Okay, then, okay, so we're gonna start off in this small neighborhood, okay? And I claim that we're gonna get within a n neighborhood when we map n, so this is like the r from the beginning. Okay, then this is gonna be one over lambda x i y i at x naught
31:27
is gonna be greater than n, plus c, plus k, plus delta. Okay, so this implies that one over lambda x i y i at x naught,
31:42
minus c, plus k, plus delta is greater than n. Okay, and f, the Gromov product of the images is bigger than this thing, okay? So if I get in a small enough neighborhood, I can get my other thing in a small enough neighborhood. That's continuous. Okay, all right.
32:04
So now we have an invariant of the group. Okay, so now, the boundary of G, G acts geometrically on x, proper hyperbolic metric space.
32:24
We have a well-defined, the boundary of G is just gonna be defined to be the boundary of any such x. Okay, so now we can go to town. So when this was figured out, I guess by Gromov,
32:41
in his article where he uses Gromov product, the people were wondering, what kind of stuff can this tell us? It actually can tell you a lot of stuff. Okay, this can tell you a lot. So let's look at some examples, some quick examples.
33:05
Okay, so examples. So the boundary of a free group, say F2, is gonna equal a Cantor set. So you can think about this if you want. All right, put a couple little exercises about this.
33:25
And actually, so you can think about sequences going this way. You can start to have these disjoint neighborhoods and then you can get disjoint and disjoint. It's really fun to think about this. And for a trivalent tree, it's a little easier if you like the middle thirds Cantor set,
33:43
which everybody is prejudiced towards. Yes, and then actually, if G is hyperbolic, boundary of G is equal to a Cantor set,
34:01
then G is virtually free. Okay, so another nice example is the boundary of some group, closed surface group, like you saw yesterday. So a closed surface group acts geometrically, eight,
34:27
tiles H2, okay? So the boundary of a closed surface group is gonna equal S1, okay, the boundary of H2. And if boundary of G equals S1 and G is hyperbolic,
34:48
then G is virtually a surface group. This is much harder. This is a goodbye, Tukya, Cass and Jungreis,
35:07
a lot of work, a lot of non-trivial work. Okay, so some open problems that you can immediately start formulating as soon as you know the definition.
35:20
So you can take your favorite boundary and try to figure out all the things that have that boundary. So problem boundary M,
35:47
which will be a, you'll need a piano continuum, have to have a lot of symmetry, okay? And which hyperbolic G have the boundary
36:02
of G homeomorphic to them? Okay, so one interesting question that you might wanna work on that's actually been resolved. So this proof about quasi-isometry shows that if the groups are quasi-isometric, then they have homeomorphic boundaries, okay?
36:22
There are examples of non-quasi-isometric groups that have homeomorphic examples, so be a little careful. Okay, that we use the quasi-conformal structure at infinity that I'm not going into, but they exist. Okay, and then you might even wonder, maybe you take your favorite space
36:40
and you wanna know is, so an easier version of this, maybe this is the empty set, is M the boundary of anything, of any hyperbolic group? So maybe, or characterize the spaces which M occur.
37:15
Okay, there's been some work done on this. Okay, those are both kinda hard, but there's things you can start thinking about.
37:23
All right, so one thing that you can see is you can see the boundaries of subgroups. So the geometry of subgroups is really interesting, and certain subgroups you can see in the boundary very nicely, so let me go over that. The geometry of subgroups can really tell you a lot about the group,
37:41
and I'm happy to go on about that later. I like thinking about geometry of subgroups, of subgroups. So some of these subgroups, you can actually see in the boundary. For example, if you have,
38:04
you can kinda think I've got a Z in here. Let me just do this example. Okay, I've got two points in the boundary that are invariant by that G. I can see the boundary of G. What's the boundary of Z?
38:25
Two points, good, and it's right there. Okay, maybe it's not distinguished from any other two points, but it is there. Might not be easy to see, but it's there. Okay, so this is an example of a quasi-convex subgroup, so let me just say what that is.
38:40
So a subset A of a geodesic matrix space is, I'll say, E quasi-convex,
39:03
and I won't care about the E, okay? If any geodesic on A2 and X,
39:25
so where A1 and A2 are in my subset A between points of A, is a bounded distance from A.
39:41
So an E, so is distance E from A, neighborhood of A. Okay, so the picture you might wanna have in mind is some kind of blob.
40:03
Okay, it might not be convex, but it's quasi-convex. So, and then I'm gonna say A, so all my quasi-convex subgroups are gonna be infinite, so A, an infinite group, subgroup, is quasi-convex
40:30
if it is quasi-convex in some Cayley graph,
40:41
so G is gonna be finitely generated, if it is quasi-convex in the Cayley graph of G, S, where S is some finite generating set.
41:01
So I'm being a little careful here, I won't have to be careful in a minute, but I haven't said anything about hyperbolic yet, so there are actually groups that can be quasi-convex in one generating set and not the other, so that's horrible, so let's go to hyperbolic groups.
41:21
So for hyperbolic groups, we can just say, oh, it is quasi-convex in some group, some space on which the group acts geometrically, which is what we like. So G is hyperbolic, doesn't depend on the S,
41:48
and I can think A, quasi-convex, because I'd rather think about just some space, my favorite space on which the group acts geometrically, if and only if the orbit of A, A X naught
42:05
is quasi-convex in X, where G acts geometrically on X, so X itself will be hyperbolic if it acts geometrically. Okay, so this is just pick some point and then move it around by your subgroup.
42:22
Okay, so let's talk about how we could get something, so now we have a hyperbolic group, we've got a wonderful boundary, we can think how is this gonna tell us about the boundary, so the limit set, so A contained in G hyperbolic, the limit set, you could talk about the limit set
42:41
of some subset of X, but let's just think about an orbit here. I'm gonna have the orbit of some point is gonna be a subset of X, okay, and then the limit set of the subgroup A is just gonna equal all the equivalence classes
43:01
of sequences that just come from that orbit. Okay, so let me just write that down, the equivalence class of sequences where Xi is somehow in this, say this orbit. Okay, so for example, yesterday you saw quasi-Fouxian group here, where you had some group,
43:23
a surface group that actually was acting on H three, but wasn't exactly like acting as PSL two Rs, I had a, was acting in PSL two C and the limit set here was still topologically a circle,
43:43
because it'll turn out that this, when the group is quasi-convex, this is a, this is gonna be a quasi-convex subset, I'll say what I mean by this, but this was the limit set of the orbit of that subgroup of PSL two C. Okay, so example, quasi-Fouxian.
44:05
Okay, and this has been used for climbing groups for a long time to understand a lot about the subgroups. Okay, so the limit set was really used in climbing groups. In fact, in my mind, the whole definition of the boundary is a generalization of the limit set of a climbing group.
44:23
Okay, so let me just say one thing. So I'm on my own timeframe, but I'll stop in like two minutes, is that okay? Okay, so let's talk about the quasi-convex subgroups.
44:53
Okay, so now I have this X, proper hyperbolic,
45:01
and I have some closed subset of the boundary. I can look at the convex whole of N, which is gonna equal the union of A, B. Now what do I mean by these A, Bs? Okay, so I wanna look at lines between points on the boundary in this set. So we're A, B, R, and N, okay?
45:22
So in a hyperbolic space, so if I have A, B contained in the boundary of X, there exists, this is geodesic space, there exists a bi-infinite geodesic. You can get these by approximating the sequences
45:43
tending to these two points. R, I, so R that takes negative infinity to infinity to X. And I want this side of the sequence to converge to A and this side of the sequence to converge to B. So I mean the limit as I goes to infinity
46:03
of R, negative I equals A. Okay, so this means that this sequence, R to the negative I, its equivalence class is A. And the limit as I goes to infinity of R of I equals B.
46:21
So I can't stop drawing H2, but it looks like this. Okay, and I have this geodesic. Okay, so to form the convex whole, I'm gonna take all of those lines between points of the boundary, okay? And we're gonna call this, so this is the convex whole.
47:02
Okay, so if I have a, so let's give some definitions for quasi-convex, did I say what a quasi-convex subgroup was on the orbit to be quasi-convex, did I say that? Okay, yeah?
47:38
In AB, do we pick choose a geodesic
47:41
or do we take all that may exist? Take all of them if you want. So there, and it's gonna be fine. The set of lines between points, okay. So just proposition about what happens with quasi-convex groups.
48:05
So if I have G acting geometrically on X, so the convex, so if I have quasi-convex subgroups of G,
48:25
of G or hyperbolic, so I like to think of this, so it turns out that the convex whole of the limit set of a quasi-convex subgroup is gonna be itself delta hyperbolic
48:40
with maybe some other delta, it's quasi-convex. And you can think of the group acting on that or you can think of the group acting on its orbit if you like, because that's quasi-convex. And that's gonna be a hyperbolic space for this subgroup and it acts geometrically on that. So, A contained in G is quasi-convex exactly when,
49:08
so I can think of this convex whole. The convex whole of the limit set of A, remember that's the limit set of things coming from the orbit all over A has finite diameter.
49:21
Okay, so this is exactly analogous to what happens in H3 where you have a limit set, you take the convex whole of the limit set, this gives you a space that it acts on, your group acts on it and downstairs you get a compact manifold. Okay, in general, we're not gonna get a manifold but we'll get some object that's compact.
49:40
Okay, so that's why the geometry of it is easier to deal with. And also we have that if A contains in G is quasi-convex, then the boundary of A is gonna embed in the boundary of G which just equals the limit set of A.
50:01
Okay, so we have these hyperbolic subgroups and we can see their boundaries in the boundary of our whole group. Okay, so, oh yeah, yeah, yeah, sorry.
50:24
So, let me just introduce, talk one minute on the next topic or I'll just stop, I can just stop right there.