Spaces of Graphs, Tori and Other Flat Gamma-complexes
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Vogtmann, Karen
Spaces of finite graphs play a key role in perturbative quantum field theory, but also in many other areas of science and mathematics. Among these is geometric group theory, where they are used to model groups of automorphism of free groups. Graphs can be thought of as 1-dimensional flat metric spaces.In higher dimensions, spaces of flat n-dimensional tori model automorphism groups of free abelian groups.There are very interesting groups which interpolate between free groups and free abelian groups, called right-angled Artin groups. I will describe a space of “Gamma-complexes”, which are a hybrid of tori and graphs, and which model automorphism groups of right-angled Artin groups, by recent joint work with Bregman and Charney.
 Well, thank you. Thanks for the invitation to speak at DERC's birthday conference. I should say that he had a preview of this talk already, probably the last, well, the second to the last time I traveled in the last year. So I went to Berlin in December of 2019. I made one other trip and since then I've never been more than a mile from my house. Anyway, so he did have a preview of this talk. I hope he'll still find something interesting in this talk. So what am I going to talk about? Many of the talks in this conference have talked about non-commutative and commutative versions of algebraic structures. And I'm normally a topologist and a group theorist and for me, the these universes, commutative and non-commutative universes, the inhabitants, the free inhabitants of these universes are free groups and free abelian groups. So in both cases, free groups and free abelian groups, the inhabitants exhibit a lot of symmetry, namely any permutation of the generators generates an automorphism of the groups. But what I want to do today is break this symmetry and talk about some sort of intermediate universe that kind of lives between the non-commutative and commutative worlds, which is I'm going to talk about partially commutative universes. So, whoops, if non-commutative universes are blue and commutative ones are yellow, then I guess partially commutative ones should be green. Right. So the inhabitants of this universe to meet for me would be, it turns out that there's a whole zoo of free inhabitants of this universe, namely groups that are free except for their partially commutative, which these days are more commonly known as rags. Rags stands for right angled art and groups and their right angle because they're closely related to right angled coxeter groups, which are reflection groups where the hyperplanes are at right angles to each other. So instead the zoo here contains right angled art and groups. Okay, so the, whoops, I forgot the right angle art and groups. So it turns out that breaking this symmetry results in really amazing, weird and wonderful things happening to these groups. In particular, they have a surprisingly rich set of subgroups. Of course, free groups, any subgroup of a free group is a free group, and commutative group, free abelian groups, any subgroup of a free abelian group is a free abelian group. In fact, it's even restricted in rank, it can't have rank bigger than its parent group. But a subgroup of a right angled art and group can be quite wild. And they've come into the focus recently because you can find every hyperbolic three manifold group has a finite index subgroup, which is a subgroup of a right angled art and group. So this was part of some very spectacular work of agel and wise that helped to settle the last of Thurston's conjectures on the structure of three manifolds. So these groups are very trendy now in group theory and in low dimensional topology. So each of these partially commutative groups, each of these rags, G has its own set of symmetries. And the way you visualize this is you draw a graph where the vertices are the generators, the free generators, and you get an edge that goes between any two vertices that commute. So you write vertices for this is a right angled art and group on five generators. And you might have maybe those commute and the others don't. So you would call this group, the associated group is called a sub gamma because you can the graph gamma completely determines the group. So notice that any automorphism of this group is going to have to preserve the commuting relations. So said in terms of this graph, it's going to have to be an automorphism of the graph. Any automorphism of the graph given by a permutation of the generators, any automorphism of the group is an automorphism of gamma. So that gives you some automorphisms of this group a gamma, but in general, there are many more automorphisms. For instance, if gamma didn't have any edges, then the automorphisms are the entire automorphism group or the free group. Right. So there's lots of automorphisms in general. Right. So that's the story with these free inhabitants of these universes, and how you describe them and what they are. I do topological and geometric group theory, so I'd like a topological model for these groups. So let's stick for the purposes of this talk to finitely generated groups. So free groups are Fn. And for as a topological model of this free group, I'll take a graph, a finite graph, and what makes it a topological model is pi one of the graph should be the free group isomorphic to the free group. For free abelian groups, on the other hand, on the other side of the spectrum, I have a natural candidate for a topological model, namely a torus of dimension n. So again, the fundamental group of the torus is isomorphic to, I'm only talking about finitely generated free abelian groups. So I get that. So I'm going to need something, if I'm going to think about this whole universe, I want to think about a topological model for partially commutative groups as well. And so those are going to be gamma complexes. So I want these things to be in particular, things that are isomorphic, whose fundamental group is the group A gamma. So I'm going to also be interested in metrics on these models. So if I take a finite graph and put lengths on this edges, I can make it into a metric space. So I'll call that a metric graph. If I take a torus over here on the right hand side, I can put in I can put lots of metrics on a torus. I don't, the nicest metrics to use are metrics that are flat. So the metric models for free abelian groups that I want to consider are going to be flat tori. And so then I have to, if I want a metric model for right angle art in groups, I need something that I'll call a flat gamma complex. Okay, so what's the point of having a metric model instead of just a topological model for these groups? Well, the point is that if I have a metric model, then for my group, then I can, I can vary the metric slightly in in Gromov's space of metric spaces. And I get another metric model. So I can make a whole space of metric models. And it turns out what's nice to do is given a metric model and an isomorphism from the fundamental group of my model to my group a gamma, I guess I've been calling these a gamma now. I can make a space by deforming the metric as I said, and this isomorphism will get pulled along with this deformation of the space. And then if I have an automorphism of the group, I get an action of the automorphism group of my group on this space, which is just changes the isomorphism. So if I have a particular point in my space, that's an isomorphism from the fundamental group of x to my group, and I take an automorphism of the group, then I compose my old isomorphism with this automorphism, I get a new isomorphism of the fundamental group of x to my to my group. So that's just written down here. The automorphism group acts on this, the space of marked metric spaces by just changing the isomorphism with the fundamental group. So the action does not change the metric. Now, so I had metric spaces. Yeah. Looks like I've just duplicated the same slide. Sorry. Okay. So I can add another row to my to my column to my table here of what belongs in the non commutative universe and what belongs in the commutative universe. I have groups, I have graphs that are topological models, I have topological models for the groups. I have metric spaces that are models for the groups, and I have a space of marked metric spaces. So it turns out that in the commutative case, the space of flat tori is something that's been studied for many, many years it's just the symmetric space SLNR mod SON in the, and it's a contractable space. It's let me morphic to a Euclidean space in fact. In the non commutative universe, there's an analog, which is called outer space. And it's a space of marked metric graphs. And the quotient by the action is just a space of graphs, the action remember just changes the isomorphism of the fundamental group with the with the model. So the quotient after if you mod out by that action, you just get unmarked graphs you get the space the modular space of graphs. And it turns out that this space is also a contractable space. The action is proper. So that means that the stabilizer of a point is a finite group. And by standard results in algebraic topology, you can the invariance of this quotient space, such as like co homology for instance, is actually an invariant of the group. You want to study invariance of the group, and your topologically or geometrically inclined. You can also study you can study this space instead. And you can, there's an interaction between the group theory and the topology and geometry that lets you go back and forth. You can figure out things about the space by knowing things about the group and you can also figure out things about the group by knowing about the space. Okay, so why am I telling you about this. This is a conference on on quantum field theory, perturbative methods in quantum field theory. Well, turns out that these spaces have variations with for graphs with leaves. And of course graphs with leaves are just Feynman diagrams. At least that's they underlie Feynman diagrams. So you can decorate them with a lot more things and you get a Feynman diagram. These spaces collect Feynman diagrams with a fixed loop order and a fixed number of external leaves in a single object, which is geometric. So, when I visited Berlin in 2015, I met, I started talking to Dirk, and told him how I think about these spaces. And he saw lots of connections between the spaces I was talking about, and perturbative quantum field theory. And I should say that he's been patiently trying to explain Feynman integrals, Kukowski rules and renormalization to me ever since. And I'm making progress. I'm not quite there yet, but I hope he hasn't lose doesn't lose patient hasn't lost patience with me yet because I think I'm making progress. But meanwhile, Dirk, and his collaborators and students have made a lot of have been exploring connections between the commonatorics and the geometry of outer space. And the tools of perturbative quantum field theory and as I just said they found connections between parametric representations of Feynman integrals and distributions on the modular space of graphs between renormalization hot valgebras and the structure at infinity of this space. And between Kukowski rules and the commonatorial structure of the partially ordered set of graphs. So, I don't know whether the partially commutative universe will also find applications in perturbative quantum field theory. But there are some I figure this is the right audience. I've noticed that, for instance, we've heard talks about relationships, but of these various hot valgebras to structures such as sub spaces of a vector space. Those are things which show up at the boundary at infinity of the commutative picture the symmetric space. So that that was in the right hand column. And, right, as I said, Durkin, his colleagues have been exploring connections between outer space and the non commutative picture. Right. So, I should say that it's, it's a, it's a good thing to do to try to peak Dirck's interest. It has many, many benefits. In addition to him finding connections between quantum field theory and outer space. Right. There's one of them. One of the benefits is pictured on your screen there. That's the view from my window at Les Usche, a couple of years ago. In the other direction. At Les Usche, I met a lot of interesting, very interesting and very nice people, many of whom are attending this conference right now. There's some you'll recognize you probably recognize this one. And right. There's one. There are many people here that you will recognize. There's also this one up here. I'm sorry Karen, I don't seem to see your, your photos. Really. Oh, what do you have? What do you see on my screen is this slide non commutative partially commutative commutative your table. That was several slides ago. Uh oh. I see your video well, and I can hear you very well. Yeah. Let me try sharing again. Yeah, maybe just reshare. Yeah. Okay. Yeah, that was different. Okay. So I think the last slide we had on the screen was when you finished the table. And to talk to about it. Well then, yeah, so this slide is when I was talking about what Dirk and his collaborators have done with these spaces. Please tell me again, please tell me if you stop seeing and I was saying that it's a good thing to get Dirk interested in what you're doing. And this is one of the benefits. Okay, this is a view from my window at Lezush. And this is another one of the benefits, which I met a lot of very interesting people. Well, there's, there's Dirk, and that's me. And here's one of the interesting people I met. This is Michi, Michi Barinski. And I found out that from this conference, I asked Michi a question that he answered a few couple months later using perturbative methods and this question answered, solved a 30 year old conjecture on the Euler characteristic of out of FN. And that there were benefits from quantum field theory to out of FN, as well as, yeah, hopefully benefits in the other direction. Okay, so that was right. So it's probably time to explain to you what these things actually are. These gamma complexes that are associated to partially commutative groups. So let's just look at some examples. If I take gamma to be have three generators, all of whom commute, then the group is Z to the n, in this case, Z cubed. So if gamma has three generators, none of which commute, then a gamma is free group on generators. Anything in between here's an example here, a and b commute B and C commute A and C don't so a, A and C generate a free group on two generators and they commute with a group generated by B. In general, if I have a graph. I got some generators that commute. And the group here is a gamma. You can't really say anything more about it in this particular case. Okay, so those are some examples. So, I want to talk now about spaces with the right fundamental group. So in the case of z cubed, I have a three tourists, s one cross s one cross s one, which is hard to draw a picture of, but instead I can draw just a three cube. I'll fill in three cube all filled in. And then I take the quotient by identifying opposite sides. What do I do for three for F three I want to take a rose. So that's a wedge product mess one, or since it's a rose I should draw it in red. Three circles joined together at a point. But I can also think of this similar late to the tourists here this this rose actually lives inside this tourists. It's just these three edges. What about the next one. This is our to cross s one. There's a two dimensional rose and a circle. A rose for the free group a circle for the Z here. So what I really have here is to Tori joined along an edge. And there's the a loop. Be loop. And this too I can I can think of it as a sub complex of this tourists. Where I just use the a B face here and the BC face and identify opposite sides. So this is supposed to give you an idea of what you do in general how you get a space with the fundamental group of partially commutative group a rag. Namely, you just take a sub complex of the the n dimensional tourists. This last example the tourists should be five dimensional so I'm not going to be able to draw it. But I just take a tourists for every complete clique. So there will be a three dimensional tourists for that clique, a three dimensional tourists for that clique, they'll intersect along a two dimensional tourists. There's another two dimensional tourists here that intersects that tourists in a circle. Okay, so so this is this complex that I've just described is called a salve d complex. And this is what I just said about it. It's the sub complex of the n dimensional tourists consisting of the K Tori spanned by the K clicks and gamma. This salve d complex is kind of the most basic example of a gamma complex. And in we call the such things cube complexes in particular. It's a cube complex because it's made of cubes glued together along faces. It can some of the faces, some of the cubes can have their faces identified with the same two faces of the same cube can be identified or or a face of a cube could be identified to a different cube. So I should say one remark here is that if I just want to get a complex with the right fundamental group, I only need to look at the two skeleton. In other words, I only need to put in a tourists for every pair of commuting generators. What's the point of putting in all of these higher dimensional Tori. Well, if I use Tori for all the K clicks I get I can guarantee that s gamma has a metric of non positive curvature. So this is a theorem of Gromov, which means that in technically the universal cover is a cat zero space. So that's the kind of metric version of non positive curvature for spaces which aren't necessarily manifolds. And the point is that geometric triangles, geodesic triangles in these spaces are thinner than at least as thin as Euclidean ones. Right. So remember what I was trying to do with these complexes. An isomorphism. Remember that an isomorphism between the fundamental group of my salvelli and my group is called a marking. And out of a gamma acts on the set of marked salvelli's by changing the marking. And this is just a remark if I don't want to specify a base point for these salvelli's salvelli's have a natural base point. But other gamma gamma complexes won't have a natural base point. And so if I don't want to specify a base point. I only get an action of the outer automorphism group. And inner automorphisms basically change the base point. So what do I want to do. I want to make a space of marked gamma complexes. I mean, I want to make a space. And I also want to prove the space has properties that are useful for group theory. And I want to make the conclusion that the salvelli's are the same. So I want to make the conclusion that the salvelli's are the same. So let me go through that briefly. Okay, so let's start with free a billion groups. So I want to make a space of marked gamma complexes. And I want to think of that Taurus remember as a metric space. And how do I think of it as a metric space. I want to think of it as Euclidean space. Modulo a lattice. So that gives my Taurus a flat metric locally it looks like Euclidean space. So I want to make a fundamental domain for a standard lattice R to the N mod Z to the N. And if I identify opposite sides I get the standard Taurus. If I take an element of the autumn outer automorphism group of Z squared, which is otherwise known as GLNZ. Then what happens to this to this Taurus will absolutely nothing happens to the Taurus. If I take that vector and that vector, they go under this element that I've chosen to that vector and that vector. But I get they generate the same lattice. So nothing happened to the lattice. What changed when I acted by my element of out of Z squared was the basis for the lattice. So, remember these. So these red and blue vectors are actually loops in the Taurus, and they give a basis for the fundamental group. So I have the same flat Taurus but I have different isomorphisms of the fundamental group with of the Taurus with Z squared. So, that's the same for us, but they have different markings. What about the free group. The, the salveady complex here is a rose. And here's an automorphism. Here's a two dimensional example to the rose with two pedals. And here's an automorphism of the free group on A and B. So, for instance, if a is this generator and B is this generator. Then this automorphism should send the red loop around a and B. And the blue loop should go to itself. And the blue loop and the red loop or a basis for the fundamental group acting by this automorphism changes the basis. I have the same metric graph. And, but I have got different loops. So I get a different isomorphism with F2. And if I, if I take a salveady and start acting by the group, I get, you know, a bunch of dots in my space, I get a discrete set, a discrete orbit of this space. How do I get from one point to another point. In either of these cases, let's start with the torii. How do I get from this guy to that guy seems obvious what I do is I start sharing the tourists. So I just skew gradually and on the way. So at the beginning at the end the Torah I have the same metric. At in the middle, these Torah are also flat Torah, but they're different metric spaces. They don't have the same shortest loops, for instance. How do you get from that rose to that rose. Well, this is a little bit maybe harder to see what I want to do is insert a new edge. And then collapse one of the old edges. So I'm going to collapse that old edge. So this is the old edge. And I'm going to, I just, it's, I put it over here. And then I'm going to collapse it and I'll get that. So it's a metric graph and I want to preserve the volume one thing. So what am I going to do. So, right. What I've shown you how to do is connect to points in this space to particular points in fact, but it turns out that if I look at the space of marked flat Torah. With volume one, I get a contractable space. On the other hand, I just showed you how to connect to roses by a graph with two vertices. And to get an actually contractable space, you need to allow graphs with more vertices. So now I can tell you what a gamma complex is in general, it's always going to be a cube complex. So that there's some way of collapsing. There's a collapsing operation which is standard in this in the theory of cube complexes called a hyperplane collapse. And that should give me back my salve. So you should be thinking about the free group case. If my group is a free group, then a gamma complex is just going to be a graph with the right fundamental group. And it's, it also the way I've defined it it won't have any separating edges so it'll be actually one particle irreducible. So I'm just collapsing operation. In this case, what am I doing I'm just collapsing a maximal tree to get back to a rose. But let me show you an example that's not of. It's not just a graph. Here's a, here's a here what the gamma complexes look like for this particular graph. So there's the salve. And there's this one, which I'm calling as gamma PI P. So collapsing any, if I collapse this central cylinder down to its waist curve. And if I collapse this circle, then I get the salve at e back. Yeah. So that's what a gamma complex is it's a cube complex. And there's a standard collapsing procedure that will collapse it back to a salve. There's a common editorial description of what a gamma complex is in terms of partitions, which I thought, listening to some of these talks I thought people might like. So, what do you do you form a graph gamma with vertices. So I start with my graph over here gamma, and I form kind of I kind of double it, I get. There's my original graph. I take new vertices for the inverses of these generators. And I connect to things if they commute but aren't inverses. So, a commutes with the inverse inverse commutes with be, etc. So that's kind of a doubled graph. And that this P that shows up in this picture, it corresponds to this edge, this extra edge that I've added in my gamma complex. And it gives me a partition into three sets that's that are determined by this new edge P. So let me draw this on the next page. Right. So how do I figure out what this partition is. Right, I had my graph here. We bring it over. Well I look at this edge P. And I notice that the front of a and the front of be are attached to one end of P. So I'll put them in one piece of the partition. Front of edge and a in the front of C. And the back of a and the back of C are at the other end of the partition. And be and be inverse or at both ends of that edge. So, I get a partition into of this, the vertices of this graph into three sets, one's called the link. And that consists of the half edges that are at both vertices. One's, the half edges at one of the vertices that just have a half edge at one vertex and one's a half edges at the other vertex. And I claim that given this partition, as long as it satisfies some basic rules I can reconstruct this space. So, here's my edge P. And I know that be and be inverse are supposed to be and be inverse are supposed to go at both edges of P. I know that a is supposed to go from one end to the other and see is supposed to go from one end to the other. And all I have to do the rules for this partition are going to allow me to fill in the rest of the picture. And I don't have the rest of the tubes in this gamma complex. And I don't have time and you probably don't have the patience to see exactly what the rules are, but they're easy to state. So, that tells you how to construct gamma complexes with two vertices. So, all you're going to need to construct gamma complexes with more vertices. And so you need a notion of compatibility of partitions. And then given any set of compatible partitions. I can construct a gamma complex. And collapsing hyperplanes corresponds to removing partitions from this description. So if I collapse them all I get back myself Eddie. Right. I have two more minutes, five more minutes. Something like that. Just a couple minutes. Anyway, what this what this looks like if my graph if my group is a free group. So, what's a partition. The, I'm not going to have my partitions the link part is always going to be empty. And I'm just going to have a partition. So, the sets of gamma that this is this is this is what the double of gamma looks like. And a partition is just going to be partition into two pieces. And two partitions are compatible. If they can be drawn by cert they can be described by circles that don't intersect. And then there's a picture of three compatible partitions. And whenever you have such a set of circles that don't intersect, there's a dual tree. With one vertex for every component of the decomposition into sets. And that forms a map the dual tree to that set of partitions is a maximal tree in a graph. To get the rest of the graph, you connect you add edges a b and c. So a goes from there to there. See goes from there to there. So that's a familiar picture I think to common a toriolists associating a maximal tree to a set of partitions and a graph to that. So, right. So I've described what a gamma complex is to get a space. I want to put metrics on these gamma complexes. So the cubes that I was describing won't be actually cubes. They'll be isometric to Euclidean parallelotopes. But in order to keep some control on this space, I'm going to I want these metrics to be flat. So locally cat zero. So I can finally tell you what my space is a point in my space. It's going to be a locally flat metric space. It's going to be isometric to some flat gamma complex. And it's going to be marked by an isomorphism between the complex and my right angled art and group. And the theorem. That I want to that I was trying to get to is that this is a contractable space and the action is proper. So it's quotient is a good model for this group of outer automorphisms in variants of this quotient or in variants of the group. And this is a very recent theorem of Corey, Breguin, Ruth, Charny and myself. It depends on an earlier theorem, which just talks about the combinatorial picture of gamma complexes. So, if you look at, if you make gamma complexes into a partially ordered set, the hyperplane collapse, then the geometric realization of that partially ordered set is as long as you restrict the kind of isomorphisms you allow is a contractable set. And this theorem, this recent theorem builds on that theorem. It turns out that we published this paper in 2017. And that wrote a paragraph at the end saying, well, we'll publish another short paper, adding all of the metric information and using arbitrary markings. And it turns out, as it says down here that adding the metric arbitrary markings and metric information was much harder than we anticipated. But we've now done it and are very happy that it works. So, there are various issues to be dealt with. We had to start while there's the originally combinatorics of the gamma complexes. There's straightening twisted gamma complexes. Then there's determining all possible decompositions of a metric space, all identifications with a gamma complex. So those all turned out to be major undertakings. So, that's pretty much all I wanted to say I presented you now with a new toy, Dirk. And so, I should play with it. Thank you. Thank you very much. Thank you, Kevin. This is the lovely final slide. And I have something to play. I promise I do. Okay, good. Great. It's something that you have to get to back to Berlin. You have to get you back to Berlin. Because I'm glad you haven't lost patience with me. I'm still learning. Okay, good. I have one question right away. So, as I understood it for each combinatorial graph, you define this particular group with a corresponding commutation and then to that you associate this geometric space. That's right. Now, if I have a graph and I look at a subgraph or an induced subgraph, the associated space sit in some nice way in the, in the gamma complex associated to the bigger graph. It does. So the first of all, the group is a called a special subgroup associated to the subgraph is called a special subgroup. And you can embed this space into the larger space. And there are actually several embeddings of this space, having to do with the fact that we're only looking at outer con outer automorphisms and conjugates of this subgroup give different copies of this subspace sitting inside the bigger space. Okay. And then the follow up question would be. So if I have a sub subgraph, and it's quotient graph. Is there a product of these two spaces sitting in the biggest is there some operatic structure that mimics the graphical relations. Yeah, that's not so clear. In the case of outer space. That structure shows up at infinity. So there's this way of boardifying the space by adding these quotient spaces and products of spaces and quotient spaces at the kind of it. Yeah, you don't compactify outer space you compactify the quotient the modular space of graphs. And by by doing this sort of contraction deletion contraction operations. And this is such a compactification operator, but also for the gamma complexes more generally. This, this space is very new. And that's one of my one one question I would love to answer. Yeah, I don't know. Are there other questions for Karen, you can type them in the chat or just speak up if you can have a question. So I was wondering if the graph theory tells you something interesting. So, for example, to your graph gamma, you can associate them independence complex or click complex. You get some topological spaces built sort of body independent sets in the graph or so. And yeah, this is in some way, answer the picture or tell you something about the group a gamma. The click complexes. Yeah, they have to do with, if you look at the universal cover of one of these gamma complexes. The cliques correspond to Euclidean subspaces flat flat Euclidean subspaces of the universal cover. And the clique complex. Well, I mean, this is, this is not really saying anything that it. It describes how these flat subspaces intercept. That's just translating the fact that they're sub their cliques in the graph into the, the geometry of the spaces. So, yeah, we did use this clique complex graph in the proof that this outer space is contractible. It helps you, it helps you kind of divorce yourself from the marking it's something that's inherent in the graph, it doesn't depend on the isomorphism, the fundamental group with the graph. So, yeah. I'm not sure I have a real answer to your question. I'm sure they're relevant right. Any graph theoretic constructions are probably are probably reflected somewhere in the geometry of the space. Yeah. Okay, thank you. So, if there are no further questions right now then let's thank Karen again for her beautiful talk.
 Well, thank you. Thanks for the invitation to speak at DERC's birthday conference. I should say that  he had a preview of this talk already, probably the last, well, the second to the last time I  traveled in the last year. So I went to Berlin in December of 2019. I made one other trip and  since then I've never been more than a mile from my house. Anyway, so he did have a preview of this  talk. I hope he'll still find something interesting in this talk. So what am I going to talk about?  Many of the talks in this conference have talked about non-commutative and commutative versions  of algebraic structures. And I'm normally a topologist and a group theorist and for me,  the these universes, commutative and non-commutative universes, the inhabitants, the free inhabitants  of these universes are free groups and free abelian groups. So in both cases, free groups and free  abelian groups, the inhabitants exhibit a lot of symmetry, namely any permutation of the generators  generates an automorphism of the groups. But what I want to do today is break this symmetry  and talk about some sort of intermediate universe that kind of lives between  the non-commutative and commutative worlds, which is I'm going to talk about partially commutative  universes. So,  whoops, if non-commutative universes are blue and commutative ones are yellow, then I guess  partially commutative ones should be green. Right. So the inhabitants of this universe to  meet for me would be, it turns out that there's a whole zoo of free inhabitants of this universe,  namely groups that are free except for their partially commutative,  which these days are more commonly known as rags. Rags stands for right angled art and groups  and their right angle because they're closely related to right angled coxeter groups,  which are reflection groups where the hyperplanes are at right angles to each other.  So instead the zoo here contains right angled art and groups.  Okay, so the, whoops, I forgot the right angle art and groups.  So it turns out that breaking this symmetry results in really amazing,  weird and wonderful things happening to these groups. In particular, they have a surprisingly  rich set of subgroups. Of course, free groups, any subgroup of a free group is a free group,  and commutative group, free abelian groups, any subgroup of a free abelian group is a  free abelian group. In fact, it's even restricted in rank, it can't have rank bigger than its  parent group. But a subgroup of a right angled art and group can be quite wild.  And they've come into the focus recently because you can find every hyperbolic three manifold group  has a finite index subgroup, which is a subgroup of a right angled art and group.  So this was part of some very spectacular work of agel and wise  that helped to settle the last of Thurston's conjectures on the structure of three manifolds.  So these groups are very trendy now in group theory and in low dimensional topology.  So each of these partially commutative groups, each of these rags, G has its own set of symmetries.  And the way you visualize this is you draw a graph where the vertices  are the generators,  the free generators, and you get an edge  that goes between any two vertices that commute.  So you write vertices for this is a right angled art and group on five generators.  And you might have maybe those commute and the others don't. So you would call this group,  the associated group is called a sub gamma because you can the graph gamma completely  determines the group. So notice that any automorphism of this group is going to have to preserve  the commuting relations. So said in terms of this graph,  it's going to have to be an automorphism of the graph.  Any automorphism of the graph given by a permutation of the generators, any automorphism of the group  is an automorphism of gamma.  So that gives you some automorphisms of this group a gamma, but in general, there are many  more automorphisms. For instance, if gamma didn't have any edges, then the automorphisms are the  entire automorphism group or the free group. Right. So there's lots of automorphisms in general.  Right. So that's the story with these free inhabitants of these universes,  and how you describe them and what they are. I do topological and geometric group theory,  so I'd like a topological model for these groups. So let's stick for the purposes of this talk to  finitely generated groups. So free groups are Fn. And for as a topological model of this free group,  I'll take a graph, a finite graph, and what makes it a topological model is pi one of the graph  should be the free group isomorphic to the free group. For free abelian groups, on the other hand,  on the other side of the spectrum, I have a natural candidate for a topological model, namely a torus  of dimension n. So again, the fundamental group of the torus is isomorphic to, I'm only talking about  finitely generated free abelian groups. So I get that. So I'm going to need something, if I'm going  to think about this whole universe, I want to think about a topological model for partially  commutative groups as well. And so those are going to be gamma complexes.  So I want these things to be in particular, things that are isomorphic, whose fundamental group is  the group A gamma. So I'm going to also be interested in metrics on these models. So  if I take a finite graph and put lengths on this edges, I can make it into a metric space.  So I'll call that a metric graph. If I take a torus over here on the right hand side,  I can put in I can put lots of metrics on a torus. I don't, the nicest metrics to use are  metrics that are flat. So the metric models for free abelian groups that I want to consider are  going to be flat tori. And so then I have to, if I want a metric model for right angle art in groups,  I need something that I'll call a flat gamma complex. Okay, so what's the point of having a metric model instead of just a topological model for these groups?  Well, the point is that if I have a metric model, then for my group, then I can, I can vary the metric slightly in in Gromov's  space of metric spaces. And I get another metric model. So I can make a whole space of metric models.  And it turns out what's nice to do is given a metric model and an isomorphism  from the fundamental group of my model to my group a gamma, I guess I've been calling these a gamma now.  I can make a space by deforming the metric as I said, and this isomorphism will get pulled along with this deformation of the space.  And then if I have an automorphism of the group, I get an action of the automorphism group of my group on this space, which is just changes the isomorphism.  So if I have a particular point in my space, that's an isomorphism from the fundamental group of x to my group, and I take an automorphism of the group, then I compose  my old isomorphism with this automorphism, I get a new isomorphism of the fundamental group of x to my to my group.  So that's just written down here. The automorphism group acts on this, the space of marked metric spaces by just changing the isomorphism with the fundamental group.  So the action does not change the metric.  Now, so I had metric spaces.  Yeah.  Looks like I've just duplicated the same slide. Sorry.  Okay. So I can add another row to my to my column to my table here of what belongs in the non commutative universe and what belongs in the commutative universe.  I have groups, I have graphs that are topological models, I have topological models for the groups. I have metric spaces that are models for the groups, and I have a space of marked metric spaces.  So it turns out that in the commutative case, the space of flat tori is something that's been studied for many, many years it's just the symmetric space SLNR mod SON in the, and it's a contractable space.  It's let me morphic to a Euclidean space in fact.  In the non commutative universe, there's an analog, which is called outer space.  And it's a space of marked metric graphs.  And the quotient by the action is just a space of graphs, the action remember just changes the isomorphism of the fundamental group with the with the model.  So the quotient after if you mod out by that action, you just get unmarked graphs you get the space the modular space of graphs.  And it turns out that this space is also a contractable space.  The action is proper.  So that means that the stabilizer of a point is a finite group.  And by standard results in algebraic topology, you can the invariance of this quotient space, such as like co homology for instance, is actually an invariant of the group.  You want to study invariance of the group, and your topologically or geometrically inclined. You can also study you can study this space instead.  And you can, there's an interaction between the group theory and the topology and geometry that lets you go back and forth.  You can figure out things about the space by knowing things about the group and you can also figure out things about the group by knowing about the space.  Okay, so why am I telling you about this. This is a conference on on quantum field theory, perturbative methods in quantum field theory.  Well, turns out that these spaces have variations with for graphs with leaves.  And of course graphs with leaves are just Feynman diagrams.  At least that's they underlie Feynman diagrams. So you can decorate them with a lot more things and you get a Feynman diagram.  These spaces collect Feynman diagrams with a fixed loop order and a fixed number of external leaves in a single object, which is geometric.  So,  when I visited Berlin in 2015, I met, I started talking to Dirk, and told him how I think about these spaces.  And he saw lots of connections between the spaces I was talking about, and perturbative quantum field theory.  And I should say that he's been patiently trying to explain Feynman integrals, Kukowski rules and renormalization to me ever since.  And I'm making progress. I'm not quite there yet, but I hope he hasn't lose doesn't lose patient hasn't lost patience with me yet because I think I'm making progress.  But meanwhile, Dirk, and his collaborators and students have made a lot of have been exploring connections between the commonatorics and the geometry of outer space.  And the tools of perturbative quantum field theory and as I just said they found connections between  parametric representations of Feynman integrals and distributions on the modular space of graphs between renormalization hot valgebras and the structure at infinity of this space.  And between Kukowski rules and the commonatorial structure of the partially ordered set of graphs.  So, I don't know whether the partially commutative universe will also find applications in perturbative quantum field theory.  But there are some I figure this is the right audience. I've noticed that, for instance, we've heard talks about relationships, but of these various hot valgebras to structures such as sub spaces of a vector space.  Those are things which show up at the boundary at infinity of the commutative picture the symmetric space. So that that was in the right hand column.  And, right, as I said, Durkin, his colleagues have been exploring connections between outer space and the non commutative picture.  Right. So, I should say that it's, it's a, it's a good thing to do to try to peak Dirck's interest. It has many, many benefits. In addition to him finding connections between quantum field theory and outer space.  Right. There's one of them. One of the benefits is pictured on your screen there. That's the view from my window at Les Usche, a couple of years ago.  In the other direction.  At Les Usche, I met a lot of interesting, very interesting and very nice people, many of whom are attending this conference right now. There's some you'll recognize you probably recognize this one.  And right.  There's one.  There are many people here that you will recognize.  There's also this one up here. I'm sorry Karen, I don't seem to see your, your photos.  Really.  Oh, what do you have? What do you see on my screen is this slide non commutative partially commutative commutative your table.  That was several slides ago.  Uh oh.  I see your video well, and I can hear you very well. Yeah.  Let me try sharing again. Yeah, maybe just reshare.  Yeah.  Okay. Yeah, that was different. Okay.  So I think the last slide we had on the screen was when you finished the table.  And to talk to about it.  Well then, yeah, so this slide is when I was talking about what Dirk and his collaborators have done with these spaces.  Please tell me again, please tell me if you stop seeing and I was saying that it's a good thing to get Dirk interested in what you're doing.  And this is one of the benefits.  Okay, this is a view from my window at Lezush.  And this is another one of the benefits, which I met a lot of very interesting people.  Well, there's, there's Dirk, and that's me.  And here's one of the interesting people I met. This is Michi, Michi Barinski.  And I found out that from this conference, I asked Michi a question that he answered a few couple months later using perturbative methods and this question answered,  solved a 30 year old conjecture on the Euler characteristic of out of FN.  And that there were benefits from quantum field theory to out of FN, as well as, yeah, hopefully benefits in the other direction.  Okay, so that was right.  So it's probably time to explain to you what these things actually are. These gamma complexes that are associated to partially commutative groups.  So let's just look at some examples. If I take gamma to be have three generators, all of whom commute, then the group is Z to the n, in this case, Z cubed.  So if gamma has three generators, none of which commute, then a gamma is free group on generators.  Anything in between here's an example here, a and b commute B and C commute A and C don't so a, A and C generate a free group on two generators and they commute with  a group generated by B.  In general, if I have a graph.  I got some generators that commute.  And the group here is a gamma.  You can't really say anything more about it in this particular case.  Okay, so those are some examples.  So, I want to talk now about spaces with the right fundamental group.  So in the case of z cubed, I have a three tourists, s one cross s one cross s one, which is hard to draw a picture of, but instead I can draw  just a three cube.  I'll fill in three cube all filled in.  And then I take the quotient by identifying opposite sides.  What do I do for  three for F three I want to take a rose.  So that's a wedge product mess one, or since it's a rose I should draw it in red.  Three circles joined together at a point.  But I can also think of this  similar late to the tourists here this this rose actually lives inside this tourists. It's just these three edges.  What about the next one.  This is our to cross s one. There's a two dimensional rose and a circle.  A rose for the free group a circle for the Z here.  So what I really have here is to Tori  joined along an edge.  And  there's the a loop.  Be loop.  And this too I can I can think of it as a sub complex of this tourists.  Where  I just use  the a B face here and the BC face  and identify opposite sides.  So this is supposed to give you an idea of what you do in general how you get a space with the fundamental group of partially commutative group a rag.  Namely, you just take  a sub complex of the  the n dimensional tourists. This last example the tourists should be five dimensional so I'm not going to be able to draw it.  But I just take a tourists for every complete clique.  So there will be a three dimensional tourists for that clique, a three dimensional tourists for that clique, they'll intersect along a two dimensional tourists.  There's another two dimensional tourists here that intersects that tourists in a circle.  Okay, so  so this is this complex that I've just described is called a salve d complex.  And this is what I just said about it.  It's the sub complex of the n dimensional tourists consisting of the K Tori spanned by the K clicks and gamma.  This salve d complex is kind of the most basic example of a gamma complex.  And in we call the such things  cube complexes in particular.  It's a cube complex because it's made of cubes glued together along faces.  It can some of the faces, some of the cubes can have their faces identified  with the same two faces of the same cube can be identified or or a face of a cube could be identified to a different cube.  So  I should  say one remark here is that  if I just want to get a complex with the right fundamental group, I only need to look at the two skeleton. In other words, I only need to put in a tourists for every pair of commuting generators.  What's the point of putting in all of these higher dimensional Tori. Well, if I use Tori for all the K clicks I get  I can guarantee that s gamma has a metric of non positive curvature.  So this is a theorem of Gromov,  which means that in technically the universal cover is a cat zero space. So that's the  kind of metric version of non positive curvature for spaces which aren't necessarily manifolds.  And  the point is that geometric triangles, geodesic triangles in these spaces are thinner than at least as thin as Euclidean ones.  Right.  So remember what I was trying to do with these complexes.  An isomorphism. Remember that an isomorphism between the fundamental group of my salvelli and my group is called a marking.  And out of a gamma acts on the set of marked salvelli's by changing the marking.  And this is just a remark if I don't want to specify a base point for these salvelli's salvelli's have a natural base point.  But other gamma gamma complexes won't have a natural base point.  And so if I don't want to specify a base point. I only get an action of the outer automorphism group.  And inner automorphisms basically change the base point.  So what do I want to do. I want to make a space of marked gamma complexes.  I mean, I want to make a space.  And I also want to prove the space has properties that are useful for group theory.  And I want to make the conclusion that the salvelli's are the same.  So I want to make the conclusion that the salvelli's are the same.  So let me go through that briefly.  Okay, so let's start with free a billion groups.  So I want to make a space of marked gamma complexes.  And I want to think of that Taurus remember as a metric space.  And how do I think of it as a metric space. I want to think of it as Euclidean space.  Modulo a lattice.  So that gives my Taurus a flat metric locally it looks like Euclidean space.  So I want to make a fundamental domain for a standard lattice R to the N mod Z to the N.  And if I identify opposite sides I get the standard Taurus.  If I take an element of the autumn outer automorphism group of Z squared, which is otherwise known as GLNZ.  Then what happens to this to this Taurus will absolutely nothing happens to the Taurus.  If I take that vector and that vector, they go under this element that I've chosen to that vector and that vector.  But I get they generate the same lattice.  So nothing happened to the lattice.  What changed when I acted by my element of out of Z squared was the basis for the lattice.  So, remember these. So these red and blue vectors are actually loops in the Taurus, and they give a basis for the fundamental group.  So I have the same flat Taurus but I have different isomorphisms of the fundamental group with of the Taurus with Z squared.  So, that's the same for us, but they have different markings.  What about the free group.  The, the salveady complex here is a rose.  And here's an automorphism. Here's a two dimensional example to the rose with two pedals.  And here's an automorphism of the free group on A and B.  So, for instance, if a is this generator and B is this generator.  Then this automorphism should send the red loop around a and B.  And the blue loop should go to itself.  And the blue loop and the red loop or a basis for the fundamental group acting by this automorphism changes the basis. I have the same metric graph.  And, but I have got different loops. So I get a different isomorphism with F2.  And if I, if I take a salveady and start acting by the group, I get, you know, a bunch of dots in my space, I get a discrete set, a discrete orbit of this space.  How do I get from one point to another point.  In either of these cases, let's start with the torii. How do I get from this guy to that guy seems obvious what I do is I start sharing the tourists.  So I just skew gradually and on the way.  So at the beginning at the end the Torah I have the same metric.  At in the middle, these Torah are also flat Torah, but they're different metric spaces.  They don't have the same shortest loops, for instance.  How do you get from that rose to that rose.  Well, this is a little bit maybe harder to see what I want to do is insert a new edge.  And then collapse one of the old edges. So I'm going to collapse that old edge.  So this is the old edge.  And I'm going to, I just, it's, I put it over here.  And then I'm going to collapse it and I'll get that.  So it's a metric graph and I want to preserve the volume one thing.  So what am I going to do. So, right.  What I've shown you how to do is connect to points in this space to particular points in fact, but it turns out that if I look at the space of marked flat Torah.  With volume one, I get a contractable space.  On the other hand, I just showed you how to connect to roses by a graph with two vertices.  And to get an actually contractable space, you need to allow graphs with more vertices.  So now I can tell you what a gamma complex is in general, it's always going to be a cube complex.  So that there's some way of collapsing. There's a collapsing operation which is standard in this in the theory of cube complexes called a hyperplane collapse.  And that should give me back my salve.  So you should be thinking about the free group case. If my group is a free group, then a gamma complex is just going to be a graph with the right fundamental group.  And it's, it also the way I've defined it it won't have any separating edges so it'll be actually one particle irreducible.  So I'm just collapsing operation. In this case, what am I doing I'm just collapsing a maximal tree to get back to a rose.  But let me show you an example that's not of.  It's not just a graph.  Here's a, here's a here what the gamma complexes look like for this particular graph.  So there's the salve.  And there's this one, which I'm calling as gamma PI P.  So collapsing any, if I collapse this central cylinder down to its waist curve.  And if I collapse this circle, then I get the salve at e back.  Yeah.  So that's what a gamma complex is it's a cube complex. And there's a standard collapsing procedure that will collapse it back to a salve.  There's a common editorial description of what a gamma complex is in terms of partitions, which I thought, listening to some of these talks I thought people might like.  So, what do you do you form a graph gamma with vertices. So I start with my graph over here gamma, and I form kind of I kind of double it, I get.  There's my original graph. I take new vertices for the inverses of these generators.  And I connect to things if they commute but aren't inverses. So, a commutes with the inverse inverse commutes with be, etc.  So that's kind of a doubled graph.  And that this P that shows up in this picture, it corresponds to this edge, this extra edge that I've added in my gamma complex.  And it gives me a partition into three sets that's that are determined by this new edge P. So let me draw this on the next page.  Right. So how do I figure out what this partition is.  Right, I had my graph here.  We bring it over.  Well I look at this edge P.  And I notice that the front of a and the front of be are attached to one end of P. So I'll put them in one piece of the partition.  Front of edge and a in the front of C.  And the back of a and the back of C are at the other end of the partition. And be and be inverse or at both ends of that edge.  So, I get a partition into of this, the vertices of this graph into three sets, one's called the link.  And that consists of the half edges that are at both vertices.  One's, the half edges at one of the vertices that just have a half edge at one vertex and one's a half edges at the other vertex.  And I claim that given this partition, as long as it satisfies some basic rules I can reconstruct this space.  So, here's my edge P.  And I know that be and be inverse are supposed to be and be inverse are supposed to go at both edges of P.  I know that a is supposed to go from one end to the other and see is supposed to go from one end to the other.  And all I have to do the rules for this partition are going to allow me to fill in the rest of the picture.  And I don't have the rest of the tubes in this gamma complex.  And I don't have time and you probably don't have the patience to see exactly what the rules are, but they're easy to state.  So, that tells you how to construct gamma complexes with two vertices.  So, all you're going to need to construct gamma complexes with more vertices.  And so you need a notion of compatibility of partitions.  And then given any set of compatible partitions.  I can construct a gamma complex.  And collapsing hyperplanes corresponds to removing partitions from this description. So if I collapse them all I get back myself Eddie.  Right.  I have  two more minutes, five more minutes.  Something like that.  Just a couple minutes. Anyway,  what this what this looks like if my graph if my group is a free group.  So, what's a partition.  The, I'm not going to have my partitions the link part is always going to be empty.  And I'm just going to have a partition.  So, the sets of gamma that this is this is this is what the double of gamma looks like.  And a partition is just going to be partition into two pieces.  And two partitions are compatible.  If they can be drawn by cert they can be described by circles that don't intersect.  And then there's a picture of three compatible partitions.  And whenever you have such a set of circles that don't intersect, there's a dual tree.  With one vertex for every component of the decomposition into sets.  And that forms a map the dual tree to that set of partitions is a maximal tree in a graph.  To get the rest of the graph, you  connect you add edges a b and c.  So a goes from  there to there.  See goes from there to there.  So that's a familiar picture I think to common  a toriolists associating a maximal tree to a set of partitions and a graph to that.  So, right.  So I've described what a gamma complex is to get a space.  I want to put metrics on these gamma complexes.  So the cubes that I was describing won't be actually cubes.  They'll be isometric to Euclidean parallelotopes.  But in order to keep some control on this space, I'm going to I want these metrics to be flat.  So locally cat zero.  So I can finally tell you what my space is a point in my space.  It's going to be a locally flat metric space.  It's going to be isometric to some flat gamma complex.  And it's going to be marked by an isomorphism between the complex and my right angled art and group.  And the theorem.  That I want to that I was trying to get to is that this is a contractable space and the action is proper.  So it's quotient is a good model for this group of outer automorphisms in variants of this quotient or in variants of the group.  And this is a very recent theorem of Corey, Breguin, Ruth, Charny and myself.  It depends on an earlier theorem, which just talks about the combinatorial picture of gamma complexes.  So, if you look at, if you make gamma complexes into a partially ordered set, the hyperplane collapse, then the geometric realization of that partially ordered set  is as long as you restrict the kind of isomorphisms you allow is a contractable set.  And this theorem, this recent theorem builds on that theorem.  It turns out that we published this paper in 2017.  And that wrote a paragraph at the end saying, well, we'll publish another short paper, adding all of the metric information and using arbitrary markings.  And it turns out, as it says down here that adding the metric arbitrary markings and metric information was much harder than we anticipated.  But we've now done it and are very happy that it works.  So, there are various issues to be dealt with.  We had to start while there's the originally combinatorics of the gamma complexes.  There's straightening twisted gamma complexes.  Then there's determining all possible decompositions of a metric space, all identifications with a gamma complex.  So those all turned out to be major undertakings.  So, that's pretty much all I wanted to say I presented you now with a new toy, Dirk.  And so, I should play with it.  Thank you.  Thank you very much. Thank you, Kevin.  This is the lovely final slide.  And I have something to play. I promise I do.  Okay, good. Great.  It's something that you have to get to back to Berlin.  You have to get you back to Berlin.  Because I'm glad you haven't lost patience with me. I'm still learning.  Okay, good.  I have one question right away.  So, as I understood it for each combinatorial graph, you define this particular group with a corresponding commutation and then to that you associate this geometric space.  That's right.  Now, if I have a graph and I look at a subgraph or an induced subgraph, the associated space sit in some nice way in the, in the gamma complex associated to the bigger graph.  It does.  So the first of all, the group is a called a special subgroup associated to the subgraph is called a special subgroup.  And you can embed this space into the larger space.  And there are actually several embeddings of this space, having to do with the fact that we're only looking at outer con outer automorphisms and conjugates of this subgroup give different copies of this subspace sitting inside the bigger space.  Okay. And then the follow up question would be.  So if I have a sub subgraph, and it's quotient graph. Is there a product of these two spaces sitting in the biggest is there some operatic structure that mimics the graphical relations.  Yeah, that's not so clear.  In the case of outer space.  That structure shows up at infinity. So there's this way of boardifying the space by adding these quotient spaces and products of spaces and quotient spaces at the kind of it.  Yeah, you don't compactify outer space you compactify the quotient the modular space of graphs.  And by by doing this sort of contraction deletion contraction operations.  And this is such a compactification operator, but also for the gamma complexes more generally.  This, this space is very new.  And that's one of my one one question I would love to answer. Yeah, I don't know.  Are there other questions for Karen, you can type them in the chat or just speak up if you can have a question.  So I was wondering if the graph theory tells you something interesting.  So, for example, to your graph gamma, you can associate them independence complex or click complex.  You get some topological spaces built sort of body independent sets in the graph or so. And yeah, this is in some way, answer the picture or tell you something about the group a gamma.  The click complexes.  Yeah, they have to do with, if you look at the universal cover of one of these gamma complexes.  The cliques correspond to Euclidean subspaces flat flat Euclidean subspaces of the universal cover.  And the clique complex.  Well, I mean, this is, this is not really saying anything that it.  It describes how these flat subspaces intercept.  That's just translating the fact that they're  sub their cliques in the graph into the, the geometry of the spaces.  So,  yeah, we did use this clique complex graph in the proof that this outer space is contractible.  It helps you, it helps you kind of divorce yourself from the marking it's something that's inherent in the graph, it doesn't depend on the isomorphism, the fundamental group with the graph.  So, yeah.  I'm not sure I have a real answer to your question. I'm sure they're relevant right. Any graph theoretic constructions are probably are probably reflected somewhere in the geometry of the space.  Yeah.  Okay, thank you.  So, if there are no further questions right now then let's thank Karen again for her beautiful talk.
Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_0_71.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_72_119.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_120_395.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_396_1403.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_1404_2495.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_2496_2891.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_2892_3155.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_3156_3239.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_3240_4511.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_4512_6191.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_6192_8327.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_8328_10655.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_10656_10931.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_10932_12371.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_12372_16859.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_16860_21011.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_21012_32747.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_32748_32795.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_32796_33035.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_33036_33371.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_33372_33635.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_33636_33695.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_33696_34223.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_34224_34451.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_34452_35003.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_35004_36035.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_36036_38315.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_38316_38363.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_38364_43679.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_43680_47303.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_47304_49223.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_49224_54803.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_54804_55451.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_55452_55511.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_55512_55691.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_55692_56447.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_56448_57551.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_57552_58403.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_58404_58487.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_58488_60287.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_60288_63971.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_63972_64295.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_64296_64475.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_64476_65807.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_65808_67403.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_67404_67523.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_67524_67715.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_67716_68447.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_68448_71699.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_71700_71999.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_72000_76199.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_76200_77327.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77328_77375.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77376_77435.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77436_77507.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77508_77555.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77556_77615.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77616_77663.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77664_77735.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77736_77783.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77784_77843.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77844_77891.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_77892_78083.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_78084_78935.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_78936_79523.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_79524_80219.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_80220_80279.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_80280_80327.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_80328_80639.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_80640_80735.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_80736_81287.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_81288_81443.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_81444_81599.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_81600_82079.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_82080_82559.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_82560_83051.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_83052_83207.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_83208_83519.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_83520_83687.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_83688_84323.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_84324_84971.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_84972_85259.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_85260_85319.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_85320_85379.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_85380_85439.png Karen_VOGTMANN_-_Spaces_of_Graphs__Tori_and_Other__-iIwr8pUtuE_85440_85454.png