Joseph A Quinn - Hilbert-Blumenthal Quaternionic Surfaces - JMM2018 Quaternion Session

Transcript

is joseph queen and joseph queen is from the Universidad Nacional autónoma adore ma de Mexico and he's going to talk about Hilbert Blumenthal cuiture neonics surfaces and here he is that first of all thanks very much so yeah I'm going to talk about something called there were looming attorney onyx surfaces and this is an idea of my collaborator and my mentor my postdoc El Greco and what it is essentially is it's a periodic generalization of the classical Hilbert Blumenthal services so the first thing I want to do is just kind of familiarize you or maybe remind you of what the classical thing is look at how this generalizes so also in the previous talk luckily we had a discussion of what manifold is and all this kind of stuff our perspectives a little bit different though it's not really from a differential TMF geometry perspective we're more interested in generalizing with types of techniques that are used to study hyperbolic circuses and 3-manifolds so we're thinking the more topological radicals especially from the perspective of what you from having a good fundamental domain in these things so the the idea of basically starts with this idea where you take the upper half-plane model for the hyperbolic plane and then I get an action by Mobius transformations of PSL 2r which is 2 by 2 matrices determinant 1 of the sign and this is the usual Mobius transformation this gives us an action of that group on the hyperbolic plane ok now in order to get a Hilbert modular group instead of using the real numbers I take the totally real number of field I look at its ring of integers and I put that into the entries in my matrices instead this gives me a Hilbert launch of a group so that'll be our notation for that and the simplest version of this is this well-known example of when my my number field assists the rational numbers the REME introduces a regular integers and I I get a discrete action group on the hyperbolic plane and I formed the modular surface right and so these show me my identifications and I'm going to skip a lot of detail really but you can just think of it like okay we go out this way we come back in this way and these two edges are identified by a translation right and there are some other things going now things get more complicated if I looked at a higher degree number fields in particular I want to focus on when I have a quadratic field so if what 1300 degree number field it's going to be dense in the real numbers so I no longer have a discrete action of the group I can recover - treatments however if I look at the action on a product of hyperbolic points we're looking at the case of degree 2 I can do this for any every degree and have that many products and then my action is just conjugated by the field of veneks of that random integer to supply to the increase of the group so in this case all it does is negate the irrational part and we did it's not a manifold Anjali orbifold it has this like a colorpoint but this is what we call a classical over Blumenthal servers right and it's a service in the sense that it's a service over the complex not very sort of like what was talked about in the previous all right so what we want to do is we want to do this over the attorney ins instead of over the complex numbers so first we need a notion of attorney on a hyperbolic for space now we all know what Nelson's concerns are here and a cool thing we can do is I get an upper half space model for hyperbolic or space by just being for turning into a positive scalar so now it's sort of like the pure convenience or the boundary of the space and I just push upwards from the scalar part I also have your attorney on a Mobius transformation on this space there's some stuff that needs to be done to make this just right but essentially it's just like the regular one only I can't write it as a fraction because they're not commutative so I just said the convention and I'll multiply my inverse is on the right so if I do this the next thing is what is a PS l2 over the paternity excellence it's sort of non-trivial to say what matrix algebra is over in on community feel great so instead of using the ordinary determinant we use this determine where the bar is concerning conjugation and the spelled is a dude on a determinate if I just do that that's not good enough it's not gonna fix the boundary of the space he's not going to fix what I'm calling hyperbolic for space there's a version of this for hyperbolic v space where if I just do that it works I'd love to say more about that but I'm not really gonna have time but I'll just say one thing what you do is you just take this and direct something with our plus and it'll give you a cross v space or another dispatch but if I want for space I have to have this condition where conjugating this matrix fixes that matrix and this gives me what I want and that group now is isometric to the full group of positively oriented isometry synthetic forces okay so now continuing with this analogy with the classical physics and let's look at what happens if I look at the number of fields of the graph suitable numbers I have a quaternion algebra over that field and said over the reals I take an order in there it's just a lattice Z lattices four-dimensional which is a ring with unity and I put that into the entries of this group instead this will give me a discrete action on the space and I will get hydraulic four orbitals so in a previous paper of my collaborator they worked out a lot of details for these two specific classical orders the Lipchitz order and the jointer and I kind of think this is sort of like the naive version of this is the right way to generalize the ring of integers but what they didn't miss papers and worked out a lot of geometric topological dynamical properties of these specific two examples with the idea that more things could be done there's a lot of other work in a similar vein using a same construction they also did the five dimensional version in that way okay so what we want to do next so this is the new part is I want to use a quadratic field okay yeah so now I have my ring of integers of the quadratic field I looked at the algebra over that and I take an order in there now this will give me a discrete action on the front two copies of hyperbolic court space basically in the same way as before I just use the del walk-ons and and everything goes through there's just a lot more numbers this is what we want to call Hilbert Blumenthal attorney onyx services I don't call them returning on a killer Blumenthal services because somebody already used that term to mean a motor mechanic way of making regular civil or criminal sources so this is just a choice of what to call it and we're also looking at the five dimensional version but this one is nice because it really has sort of a sensible attorney on it dimension like two-dimensional over their attorney ins a real dimensions and so what do is find the fundamental domain for these types of orbitals that allows us to study their geometry topology and dynamical systems on the things so the natural thing to do is just go back we'll get everything that was done in the classical setting and try to generalize it but we found that sort of unsatisfying to the type of stuff we were interested in like my training is more in hyperbolic 3-manifolds apology Alberto Alberto knows everything but we had to feel to work together and the work on the classical Hilbert Blumenthal services is very much in the main of number theory and so we found it necessary to write our first paper on finding it nice or fundamental though would generalize better for the type of stuff we're interested in the classical piece so I'm gonna spend some time talking about that but let's keep in mind that the idea is that all this stuff is going to generalize and some parts would be the same some parts will be different okay so let's focus on the case where the warm foam has one cusp which is to say to feel this bus under one other words if it there's only one orbit on the boundary in this case I can write my my group as the stabiliser of the pair of points at infinity in my two copies of a hydrocolloid for space and this other element I have to tell you what these things are but the thing that's very convenient for making fundamental domain is that this cusp group has a semi direct product decomposition into what's like a group of parabolic isometry generalizing well from what we know about for hydraulically manifolds and a group of hyperbolic geometry so what this is doing it's like a pair of upon translations where it moves one way and one thing and then if it has an irrational part of it it's going to move differently in the other factor in this group that I'm calling de like for diagonal U is kind of like four didn't know I forget anyway so this is sort of like scaling up in one factor and down in the other okay and then these two groups send me direct product gives me my customer I want the full group I just include the pair of inversions over the pair of unit hemispheres that looks like there is something else down there but yeah [Music] there doesn't look like you're losing something on the bottom now it's definitely on my laptop actually I haven't you know yeah right okay so the idea is that if I find a fundamental domain of the region for these three groups I could intersect them into a fundamental domain fill the whole thing so some parts of this would be like from the classical version so the group D this was my pair of like scalings it makes this nice fundamental domain I'm introducing some notation up there where basically you just want to think of like take a horizontal line in each copy it's not alignments it's a three-dimensional space but you know I'm in the classical sailing sorry take a wine in each w your hyperbolic claim and then any and each one of those like cross them and form a plane now as I vary my choices of wines I just have parents of positive integers so it's like in upper quadrant so this pie slice makes a nice fundamental domain for the action of that and then you're prepared for all that this countries and right so it's going down by the fundamental unit for the negative two on one side and up by two on the other we're just moving everything over to one side for reasons that will be more clear later okay before I say anything else about that notice I skipped you that's because the group used the part we want to change the way we're modeling a fundamental domain for parabolic isometries so in the classical setting that they do is they just take and every one of these points there's a plane essentially so what they do is they just put the same tourists sort of like the canonical standard tourists at every single one of these points right and that's the part that we didn't like so I'll come back to that the fundamental domain for the inversion isometry is also fine for the classical setting we're just taking the exterior of the pair of hemispheres of half circles as far as the group you this was the topic of our first paper essentially if you use a zero slay domain for you so now I'm simply stating the same tourists at every point I'm actually respecting the changes in the metric as the point varies and what you find is that the tourists will sort of stretch an elongate but at every cross section it is it's either a rectangular or hexagonal lattice and it's very efficient algorithmically to just take the fundamental unit figure out exactly which rectangular lattices you have to connect it these fingers so what we're seeing here is a fundamental domain for the cuffs for some for the first four cases and each one of these is crossed with our plus so this is really telling us everything we need to know about the structure the other factor just changes the scale of it and it's a torus going this way and then if I go out that way what I actually have is some sort of a nossa of the do-more cos and that's gonna take me back down to the top these are solved three manifolds and you know that was known if you ask people that work on on these kind of things we already know about the solid and ice of the feel more business but you cannot find it written down anywhere in the literature so that was kind of unsatisfying to me in this approach you could actually read it down these perhaps were made using this map from the parapet well acquaintances are three where we scale the critical factor logarithmically and the Inasa of this is immersiveness precisely this has this really canonical nice way of being written down okay so how does this generalize okay all right so there's something else we said all right so if I want to look at the full group now we just looked at the cusp what about when I include this inversion isometry so this was the main problem we had with the classical construction it's that these things have finite volume they should have finite volume boundaries but if you just take the same tourists at every single one of these things instead of one of these shapes I just have like this cylinder over the torus the volume is infinite dimensional unless N equals five if you do it this way the volume is finite dimensional for N equals to 3 5 8 13 and additionally they have all these other techniques there's a bunch of work by Harvick on the 60s and 70s where he introduces this notion of a floor to control how to visualize the boundary he's very interested in a more intuitive romance through these things like we were but you don't really need this anymore because we've captured the structure in this nice way so what you're saying is you go from left to right is what happens as I take one of these lines of tour I so that from from my other pie slice thing they could like the hyperbola and that is going up the torus and then as I slide that down towards the boundary the part that's the fundamental domain for the inversion starts eating this up and I can actually piece it together as three dimensional boundary from cross-sections with these surfaces that intercept this shape okay so right the whole idea was to generalize this and this part is sort of stuff that we have yet to flesh out but this is the idea so now we're back into the case where a hyperbolic for space I haven't ordered a quadratic attorney Ellenberger quadratic field I take my discrete action on the product of copies and spaces like that of that group and I have a very similar kind of structure where the cusp group still has this semi direct product decomposition it's just a set of having an integer up here I have a pure quaternion and so acting on this space it's sort of like a six dimensional at fine group because he have these three things from the Praetorians and then each one of them has has two of linear and directions you're being Koko from a quadratic field this group of diagonal matrices I still have this fundamental unit telling me a lot but there's also a unit in the order in the quaternion so that's sort of like a rotation right so that's something that's new that has to be dealt with the rest of it generalizes very well so if I just look at the torsion free part of the cusp it's basically the same thing only instead of our cluster process all three mountain fold its r+ across the cell 7 if I want the full group again it's very similar but now my determinant is different so this takes some getting used to to think of this as being improper in person isometry but it turns out that it is and so really the main difference is this guy this this you right