Quasi-parabolic structures on groups
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Balasubramanya, Sahana
The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin. However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions remain unanswered. In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic structures and thus answer some of the open questions. I will end my talk by discussing ongoing work in the area. This talk contains some joint work with C.Abbott, D.Osin and A.Rasmussen.
 So I'd like to start by thanking the organizers for giving me the opportunity to talk about my work at this virtual conference. And I'd like to mention that part of the work that I'm talking about today is joined with Dennis Ossin, Carolyn Abbott, and some ongoing work with Alex Rasmussen. So in the area of geometric group theory, one of the tools that's available to us in order to try and study a group is to do so by means of a group action. And for any group, there's a very natural way to do this. So you start with your favorite group and you pick a generating set for your group, which for my purposes does not necessarily have to be finite. And we construct the Cayley graph of our group with respect to this generating set. And the nice thing about constructing this Cayley graph is that we've turned our group into a metric space with respect to the word metric on X. And our group has an action on this graph, which is simultaneously isometric and co-bounded. So not only do we have a metric space, it comes equipped with a natural interaction with the group. But this process is very much dependent on the choice of your generating set X. Because if you pick different generating sets, then the resulting Cayley graphs can look very different. So as an example, if our group is the set of integers and my generating set is sort of the standard generating set plus minus one, then the Cayley graph looks like a by infinite line. And this is a very intuitive way to think of the integers. But on the other hand, if I chose the generating set, which contained every element of my group, which is sort of the largest generating set I could choose, then the Cayley graph is significantly different. And it looks like a bounded set of diameter one, because any two integers are now connected by an edge, which is labeled by their difference. And we start to see that the first Cayley graph, which is the line, is very useful for the purpose of studying our group. It remembers a lot of properties of our group. But this other Cayley graph remembers nothing about the inherent structure of our group. And in fact, if I were to do this for any group and take the generating set that contained every element of my group, this is exactly the Cayley graph that I would get, it remembers nothing of the properties of the group from which it came. So at the beginning of joint work with Dennis and Carolyn, this was an observation that we wanted to formalize. We wanted to understand that when we have two generating sets, how do we say that one is better than the other for the purpose of studying our group via the associated Cayley graph? And the definition from our work is the following. So we take your generating sets x and y, and we say that x is dominated by y if the following happens. You take every element of the generating set y, and you consider its word length with respect to the other generating set, which is x, and then you take a supremum over all of these word lengths. As long as that's finite, I'm going to say that x is dominated by y. And this dominance relation is a pre-order relation. And every pre-order gives rise to an equivalence relation in exactly the same way. So we say that x and y are equivalent as generating sets if they each dominate the other. And the natural thing to do once you have an equivalence relation is to collect the equivalence class of any generating set, which we'll denote by the use of these square brackets. So what are some things to note about this dominance relation? Well, the first thing to note is that it is inclusion reversing. So the larger my generating set, the smaller it is in this dominance relation. And this is consistent with the observation that we just made, that if you choose the largest generating set, the one that contains every element of your group, it's the worst calligraph to be looking at because it doesn't remember much about your group. The second thing is that this dominance relation can be extended to the equivalence classes themselves. So I'm going to say that the equivalence class of x is dominated by the equivalence class of y if x itself is dominated by y. Or alternatively, you can find representatives in these equivalence classes that satisfy the same dominance relation. If g happens to be a finitely generated group, then the equivalence class of that finite generating set will be the largest equivalence class. And this is simply because the supremum turns into a maximum, which is always finite for a finite generating set. And the last thing to note is that if x and y are equivalent as generating sets, then it follows from my definition that the Cayley-Graphwoods respect to x and the Cayley-Graphwoods respect to y are quasi-isometric. So first of all, what is quasi-isometric? Well, it's a coarse version of isometry for spaces. And what that means is that up to having one constant that allows you to scale the metric and having potentially another constant, which you can add or subtract to the metric, these two spaces look alike. So in other words, the large-scale geometry of these spaces is the same. And from our definition, it's easy to see why this would be true, because the constants that give you the supremum in both cases will precisely determine the constants of this quasi-isometric. But I'm not just interested in equivalence classes of generating sets. I'm interested in a very specific type of equivalence class. And that is what we call a hyperbolic structure on our group. And it's the equivalence class of the generating set so that the associated Cayley-Graph is hyperbolic. And when I say hyperbolic, I mean it is a groma of hyperbolic space. So geodesic triangles are delta-thin for some delta positive. And it's OK for me to call a structure hyperbolic, because we know that hyperbolicity is a quasi-isometric invariant. So no matter which representative I pick from this equivalence class, the associated Cayley-Graph will still be a hyperbolic space. We might change the delta that's needed to make this Cayley-Graph hyperbolic, but that's OK. And I've collected all of these hyperbolic structures together into this set, which we denote by script H of G. And I'm going to endow it with the order that's induced from the dominance relation. So we have a partially ordered set. And one thing I'd like to mention about H of G at this point of time is that we don't actually only have to think of equivalence classes of generating sets. We can start in a much more general setting that of co-bounded actions of our group on hyperbolic spaces. And there is a way to put these into one-to-one correspondence with the elements of H of G. And the proof of this is really very similar to the proof of the Miller-Schwarz lemma. The primary difference is because these actions need not be proper. I cannot guarantee that the generating set that I end up with is finite. And that's why we don't request that we only start with finite generating sets to begin with. But this is just something to keep in mind. It's not particularly important for the purpose of this talk. So what are some of the things that we can say about this post-set H of G? Well, one of the things that we can say almost immediately is that it breaks down into the disjoint union of these four smaller partially ordered sets. And this follows almost immediately from the work of Gromov. So in Gromov's work, you have a group G that's acting on a hyperbolic space X. And our hyperbolic space comes equipped with a boundary. And we consider the limit points of the action of our group on the boundary of this space. And we know that this action can therefore look like one of five different kinds of actions which are essentially characterized by what this set of limit points looks like. So the first possibility is that our action is elliptic. And H of E is the post-set which contains the elliptic structures. And what this means is that there are no limit points of our group on the boundary of our hyperbolic space. And in terms of structures, it means that the elliptic structure, which is the equivalence class of the generating set that contains every element of your group, is the smallest structure. And in fact, the associated telegraph is quasi-asymetric to a point. The second possibility is that the action is linear. And H sub L is where we collect all of our linear structures. And what this means is that there are exactly two limit points on the boundary. Moreover, it also means that our group contains a loxodromic element. And by that, I mean that there is a bi-infinite axis in our space. And this loxodromic element acts along this axis by translation. And if I pick any other loxodromic any element in this setting, its limit points on the boundary are the same as this other loxodromic element. The third possibility is the one that I'm interested in the most, which is where we collect quasi-parabolic structures. And what's a quasi-parabolic action? Well, it's where we have infinitely many limit points on the boundary of our hyperbolic space. But moreover, we request that there is a global fixed point on the boundary. And the last one is where we have our general type structures. And in this case, we also have an infinite number of limit points on the boundary of our hyperbolic space. But the group has no fixed points on the boundary. So the standard example to think of general type actions are free groups that act on the heligraph that looks like an infinite tree, so for example, F2, which is the free group on two generators, acting on the heligraph that looks like a four valence tree. Now Gromo's work has one more possibility. It contains the possibility that a group acting on a hyperbolic space can be parabolic. However, parabolic actions are never co-bounded. And because I'm dealing with actions on heligraphs, those actions do not turn up in this partially ordered set. And that's why I only have four substructures here and not five. And why am I interested in H of G? Well, it gives us a way of studying all possible co-bounded actions of our group simultaneously. And the hope is that if I can understand each of these smaller partially ordered sets, then there's a way to put that information together to try and understand H of G as a whole. And so not only can I look at all the different actions of my group on hyperbolic spaces, there might be from this examination a candidate for a best action to study in order to try and understand my group via a group action. And that brings me to HQ of P, which is the post set containing the quasi parabolic structures. And the reason I'm particularly interested in them is because in our joint work with Dennis and Carolyn, we could say a lot of things about linear structures and general type structures. But there were not a lot of things that we could say about HQP of G. And that's why my interest lies in this post set. Furthermore, it was clear to us that quasi parabolic structures do behave very differently from some properties that we were able to determine for linear and general type structures. And in order to illustrate that, I'm going to cite a few theorems from our work. So it turns out that we can construct groups Gn, which have as many linear type structures as we want, but they have absolutely no quasi parabolic or general type actions on hyperbolic spaces. And we can do the same for general type structures. We can create groups Hn that have exactly n general type structures, but no quasi parabolic or linear structures. And the natural question to ask is, can we do the same for quasi parabolic structures? And the answer is no. Because if you have a quasi parabolic structure on your group, you most definitely will have a linear structure on your group, and they will be related in the sense that the linear structure is dominated by the quasi parabolic structure. And the machinery to do this comes from using the Busaman pseudo character. So a pseudo character is an almost homomorphism. So it's a homomorphism up to some bounded defect. And what we do is that we start with our quasi parabolic structure. We look at the action, the quasi parabolic action of the group on this Cayley graph. And when you run the Busaman pseudo character on it, what it does is create a projection to a line. And it does this by crushing enough of the action out in a manner that is analogous to expanding your generating set. And because of that, we get a linear structure. And because we've expanded our generating set, and our dominance relation is inclusion reversing, we get that the linear action is smaller than the quasi parabolic action. Moreover, while we had lots of examples of groups that have different linear and general type structures, we had very few examples of groups which admit quasi parabolic structures. And one of the main ones in our paper was this group Z reads Z, where you can get an anti chain of quasi parabolic structures on this group. The way to do this is to factor through the action of ZN reads Z, which are the line fighter groups acting on their bus their tree. And if you order these by divisibility, then you can get an anti chain of quasi parabolic structures here. But we were unable to come up with examples of groups that have a finite number of quasi parabolic structures. So some open questions from our paper were the following. Does there even exist a group that has a non zero finite number of quasi parabolic structures? Is there a group where we can find an uncountable chain of quasi parabolic structures? And if we request a more complicated example, is there a group where you can simultaneously have a chain and anti chain of quasi parabolic structures? And I'd like to mention that instead of quasi parabolic structures, if we ask the question for general type structures, then the answer is yes. In fact, you can pick your favorite a cylindrically hyperbolic group and HGT of that group will always contain a chain and an anti chain of cardinality continuum. So this makes it a natural question to ask in the other situation when you have an infinite number of limit points on the boundary of your hyperbolic space. And because we know that there is some relationship between linear and quasi parabolic actions, it's natural to ask that if our group does have quasi parabolic actions, then is the number of linear structures always less than or equal to the number of quasi parabolic structures? In other words, is it always true that the linear action on such a group has to come from applying the boseman pseudo character to some quasi parabolic structure? And I was able to answer these questions subsequently after our joint work with Dennis and Carolyn was done. So here are the answers to these questions. The lampliter groups, all of them, Zn reads Z for n at least two, have a finite number of quasi parabolic structures. There is a group, namely F2 reads product Z for which you can have an uncountable chain and an uncountable anti chain of quasi parabolic structures. And moreover, there exists a group where the number of linear structures strictly is larger than the number of quasi parabolic structures, which is non zero. And all three of these theorems are actually the consequence of a much more general theorem that I proved for groups that look like G reads Z. So here is that theorem. We let G be a group. And I claim that if I consider the hyperbolic structures of G reads Z, then I can always find a substructure, which I call B of G sitting inside of it that embeds into it, in fact. And what does B of G look like? B of G has this structure. So I consider the poset of proper subgroups of G ordered by inclusion, and I have these two disjoint copies of S of G. And the reason they're disjoint is because every element of this copy is going to be incomparable as a structure to every element of this copy and vice versa. However, every structure from these two copies of S of G dominates this common linear structure, and that of course dominates the smallest structure, which is the action on a point. And the two copies of SG correspond to quasi parabolic structures on our reads product. Moreover, if G is a finite cyclic group of order n, then this embedding is an equality. So for the lamplighter groups, this is exactly the structure of the hyperbolic structures on our group. And how do we prove this theorem? Well, my work heavily relies on the work of Capras, Cornulia, Mono, and Tessera. And they have this machinery of strictly confining automorphisms, which can be applied to groups that have this semi direct structure. So you have a semi direct product with Z. And I'm not going to define what a strictly confining automorphism is because it's a bit of a technical definition. But what's important for the purpose of this talk is that if you start with a subset of your base screw bay, that's strictly confining under either of the generators of Z, so T or T inverse. And from this information, we can extract a quasi parabolic structure on our group H. And it is exactly the Kruglitz class of the generating set that contains this subset of A along with T to the plus minus one. And what does the quantifier regular mean? Well, that means that if I take this quasi parabolic structure and I apply the Busaman pseudo character to it, the linear structure that I get is actually the projection of this group to the Z factor. So the homomorphism that you end up with is the projection to Z. And how does this apply to the group G read Z? Well, we can write our read product as a semi direct product where the base group is the direct sum of Z many copies of G. And how does T act on it? Well, T will take this infinite tuple and shift every element one space to the right. And T inverse will take the same tuple and shift every element one space to the left. And now I pick my subgroup H of G. And I'm going to define sets which are subsets of A that are going to be confining under the action of T and T inverse. So I define Q H, which has this structure. And I think of this as starting at index zero. I am allowed to have elements of my full group. But on all the negative indices, I only allow myself to put in elements of H. And Q prime of H is going to be the mirror image. So now starting from index zero all the way to the left, I have the full group. But on the positive indices, I only allow elements of H. And the thing to notice that this structure is compatible with the action of T because I can shift the full group. I can shift the subgroup H into the full group, but I may not necessarily be able to shift backwards. And here this structure is compatible with the action of T inverse, but not T for exactly the same reason. And it turns out that Q H is actually strictly confining under T. And Q prime of H is strictly confining under T inverse. And when I do this process for every proper subgroup H of G, what I do is that I get an entire collection of Q H's, all of which are confining under T. And I get another collection Q prime of H, all of which are confining under T inverse. And this is exactly what creates those two disjoint copies of S sub G in my partially ordered set, one that corresponds to the Q H's and the other that corresponds to the Q prime of H's. And because of the incompatibility of the action of T inverse with this structure, and similarly T with this structure, this is also intuitively what makes them incompatible to each other in our post set. And what does the regularity condition give us? Well, it's important because it gives us the common linear structure. Because no matter which one of these quasi-parable structures that I pick, and I apply the Busavan pseudo character to it, I get exactly the same homomorphism to Z. So the linear structure that they dominate from this construction is the same. The hard part of the proof is showing that there is an equality when G is a finite cyclic group of order N. And the reason is that you first have to argue that in this case, every quasi-parabolic structure firstly comes from a confining subset. And then you have to use long algebraic arguments to show that that confining subset has to be the look like QH or Q prime of H for some proper subgroup H of G. But how does this buy us the examples that we want? Well, for the land-cliar groups, this is obvious because I have an equality. And why do I have a finite number of quasi-parabolic structures? Well, because Zn has only a finite number of subgroups. And how many quasi-parabolic structures do I have? It's twice the number of proper subgroups of Zn. So that's a finite number. And for the example, which is F2 reads Z, here's what we do. We start with P of N. And this is the poset of the subsets of the natural numbers ordered by inclusion. And I embed this poset into the proper subgroups of F infinity. So F infinity here is the free group on countably infinitely many generators. And I start by indexing these generators. And for any subset of the naturals, I map it to the subgroup that's generated by the elements that have those specific indices. And now you can embed F infinity into F2, which is the free group on two generators. So you get an embedding to the proper subgroups of F2, and by the theorem, this creates an embedding into the quasi-parabolic structures of F2 reads Z. Now of course, I could have done this with if F infinity reads Z as well. The reason I wanted F2 was because I wanted an example of a group that's finitely generated with this property. And lastly, what is the example of the group that has strictly more number of linear structures? than quasi-parabolic structures? I claim it's the group that looks like this. It's the standard land fighter group direct product with Z. This group only has two quasi-parabolic structures. And those are precisely the quasi-parabolic structures that come from the land fighter group. The direct product with Z does not contribute any new quasi-parabolic structures. However, I claim that this group has uncountably many linear structures. And how do we get this? Well, our group K has two independent projections to Z. One comes from the Z factor of the wreath product, and the other comes from the projection to this copy of Z. So you can actually get a projection to Z cross Z. And now you pick your favorite line in the Euclidean plane of whatever slope you want, and I can project the entire grid down to this line. And if I do this for lines of different slope, I get very different, in fact, incomparable linear actions. And because you have a continuum worth of slopes that you can choose, you get uncountably many incomparable linear actions. And that's why this is cardinality continuum. So not only is the cardinality of the linear actions strictly greater, this is as different as it can get. So after I did this work corresponding to wreath products, Carolyn and Alex took this machinery of confining subsets, and they applied it to other groups. And the work of theirs that I want to mention here is the results that they have firstly for bump-stalked solitaire groups BS1N. These groups also have a semi-direct product structure, which looks like Z1N, semi-direct product with Z. And they were able to show that the hyperbolic structures of this group have this partially ordered set structure. So what's happening here? Well, we take the prime decomposition of N, and we assume that there are K different primes involved in the prime decomposition event, and this is that K. And then you take the power set of the set 1 up to K. That creates an embedding of this lattice here, and all of the structures here except the smallest one, which is the linear action, correspond to quasi-powerballic structures. In addition to these, there's one more that comes from the action on a hyperbolic plane. And of course, then you have the linear structure dominating the action on a point. But all of these quasi-powerballic structures also have interpretations in terms of arc and finding subsets. And the other result of theirs that I'd like to mention is the classification of hyperbolic structures on the group Z squared, semi-direct product Z, where this extension is defined by a matrix from SL2Z. So it's a two by two matrix of determinant one. And it turns out this is the structure we get. So we have two quasi-powerballic structures, which dominate a common linear structure, and that dominates the action on a point. But in terms of confining subsets, what are these quasi-powerballic structures? So we take this matrix phi and we consider it's attracting and repelling eigenline. And in terms of confining subsets of Z squared, these are neighborhoods of the attracting and the repelling eigenline for phi. So together with Alex and Carolyn, we're currently working. And what are some of the questions we're interested in? Well, we want to know, does there exist a group that has an odd number of quasi-powerballic structures? Because all of the examples we know so far have an even number. For the land-flatter groups, it's twice the number of proper subgroups of Zn. That's an even number. In the bombstock solitaire group case, it's also an even number. And for Z squared, semi-direct product with Z, it's still an even number. So does there exist a group that has an odd number of quasi-powerballic structures? And in particular, does there exist a group that has only one? Because what we'd like to do is construct groups KN that have as many quasi-powerballic structures as we want. And I'd like to mention that if I can answer this second question, then it would be immediately possible to answer the third question by taking the n-fold direct product of this group. Because for that group, the quasi-powerballic action would have to factor through one of the copies of your group, and that is exactly what will give you exactly n. And I'm also interested in understanding from the setting of the VEET product for which groups the equality holds. So I know it's true for the finite cyclic groups, but I also know it's not true in general. And I also know that it's not true for every finite group. There is a counter example in the case G is a Z2 direct product Z2. So that's the finite cyclic group of order two direct product with itself. But I suspect that this equality could hold at least for some of the finite symmetric groups. So what conditions does your group G need to ensure that this equality holds? And the work that we're focusing on is aimed at understanding the following. We'd like to classify the structures on groups that look like Z to the n semi-direct product with Z, where the extension is defined by SLNZ. And what we suspect is that in this case, as long as you consider the eigenspaces of a fee, your quasi-parabolic structures at least on this group should somehow be in some one-to-one correspondence with those eigenspaces. And what we'd like to do is also understand structures on iterated H&N extensions, with the ultimate goal being to try and understand the structures of polycyclic groups. But because these groups are not going to admit general type actions, the hope is that maybe we can find specific examples that answer these questions in this class of groups without having general type actions. Because we know that probably we won't be able to control the number of linear structures. But at least if we can have no general type actions and have as many quasi-parabolic structures as we want, then we're closer to understanding H of G as a whole. So that's sort of the direction in which we're headed. And that's all I had to say. Thank you very much.
 So I'd like to start by thanking the organizers for giving me the opportunity to talk about  my work at this virtual conference.  And I'd like to mention that part of the work that I'm talking about today is joined  with Dennis Ossin, Carolyn Abbott, and some ongoing work with Alex Rasmussen.  So in the area of geometric group theory, one of the tools that's available to us in  order to try and study a group is to do so by means of a group action.  And for any group, there's a very natural way to do this.  So you start with your favorite group and you pick a generating set for your group, which  for my purposes does not necessarily have to be finite.  And we construct the Cayley graph of our group with respect to this generating set.  And the nice thing about constructing this Cayley graph is that we've turned our group  into a metric space with respect to the word metric on X.  And our group has an action on this graph, which is simultaneously isometric and co-bounded.  So not only do we have a metric space, it comes equipped with a natural interaction  with the group.  But this process is very much dependent on the choice of your generating set X.  Because if you pick different generating sets, then the resulting Cayley graphs can look  very different.  So as an example, if our group is the set of integers and my generating set is sort  of the standard generating set plus minus one, then the Cayley graph looks like a by  infinite line.  And this is a very intuitive way to think of the integers.  But on the other hand, if I chose the generating set, which contained every element of my group,  which is sort of the largest generating set I could choose, then the Cayley graph is significantly  different.  And it looks like a bounded set of diameter one, because any two integers are now connected  by an edge, which is labeled by their difference.  And we start to see that the first Cayley graph, which is the line, is very useful for  the purpose of studying our group.  It remembers a lot of properties of our group.  But this other Cayley graph remembers nothing about the inherent structure of our group.  And in fact, if I were to do this for any group and take the generating set that contained  every element of my group, this is exactly the Cayley graph that I would get, it remembers  nothing of the properties of the group from which it came.  So at the beginning of joint work with Dennis and Carolyn, this was an observation that  we wanted to formalize.  We wanted to understand that when we have two generating sets, how do we say that one  is better than the other for the purpose of studying our group via the associated Cayley  graph?  And the definition from our work is the following.  So we take your generating sets x and y, and we say that x is dominated by y if the following  happens.  You take every element of the generating set y, and you consider its word length with respect  to the other generating set, which is x, and then you take a supremum over all of these  word lengths.  As long as that's finite, I'm going to say that x is dominated by y.  And this dominance relation is a pre-order relation.  And every pre-order gives rise to an equivalence relation in exactly the same way.  So we say that x and y are equivalent as generating sets if they each dominate the other.  And the natural thing to do once you have an equivalence relation is to collect the  equivalence class of any generating set, which we'll denote by the use of these square brackets.  So what are some things to note about this dominance relation?  Well, the first thing to note is that it is inclusion reversing.  So the larger my generating set, the smaller it is in this dominance relation.  And this is consistent with the observation that we just made, that if you choose the  largest generating set, the one that contains every element of your group, it's the worst  calligraph to be looking at because it doesn't remember much about your group.  The second thing is that this dominance relation can be extended to the equivalence classes  themselves.  So I'm going to say that the equivalence class of x is dominated by the equivalence  class of y if x itself is dominated by y.  Or alternatively, you can find representatives in these equivalence classes that satisfy  the same dominance relation.  If g happens to be a finitely generated group, then the equivalence class of that finite  generating set will be the largest equivalence class.  And this is simply because the supremum turns into a maximum, which is always finite for  a finite generating set.  And the last thing to note is that if x and y are equivalent as generating sets, then  it follows from my definition that the Cayley-Graphwoods respect to x and the Cayley-Graphwoods respect  to y are quasi-isometric.  So first of all, what is quasi-isometric?  Well, it's a coarse version of isometry for spaces.  And what that means is that up to having one constant that allows you to scale the metric  and having potentially another constant, which you can add or subtract to the metric, these  two spaces look alike.  So in other words, the large-scale geometry of these spaces is the same.  And from our definition, it's easy to see why this would be true, because the constants  that give you the supremum in both cases will precisely determine the constants of this quasi-isometric.  But I'm not just interested in equivalence classes of generating sets.  I'm interested in a very specific type of equivalence class.  And that is what we call a hyperbolic structure on our group.  And it's the equivalence class of the generating set so that the associated Cayley-Graph is  hyperbolic.  And when I say hyperbolic, I mean it is a groma of hyperbolic space.  So geodesic triangles are delta-thin for some delta positive.  And it's OK for me to call a structure hyperbolic, because we know that hyperbolicity is a quasi-isometric  invariant.  So no matter which representative I pick from this equivalence class, the associated Cayley-Graph  will still be a hyperbolic space.  We might change the delta that's needed to make this Cayley-Graph hyperbolic, but that's  OK.  And I've collected all of these hyperbolic structures together into this set, which we  denote by script H of G. And I'm going to endow it with the order that's induced from  the dominance relation.  So we have a partially ordered set.  And one thing I'd like to mention about H of G at this point of time is that we don't  actually only have to think of equivalence classes of generating sets.  We can start in a much more general setting that of co-bounded actions of our group on  hyperbolic spaces.  And there is a way to put these into one-to-one correspondence with the elements of H of G.  And the proof of this is really very similar to the proof of the Miller-Schwarz lemma.  The primary difference is because these actions need not be proper.  I cannot guarantee that the generating set that I end up with is finite.  And that's why we don't request that we only start with finite generating sets to begin  with.  But this is just something to keep in mind.  It's not particularly important for the purpose of this talk.  So what are some of the things that we can say about this post-set H of G?  Well, one of the things that we can say almost immediately is that it breaks down into the  disjoint union of these four smaller partially ordered sets.  And this follows almost immediately from the work of Gromov.  So in Gromov's work, you have a group G that's acting on a hyperbolic space X. And our hyperbolic  space comes equipped with a boundary.  And we consider the limit points of the action of our group on the boundary of this space.  And we know that this action can therefore look like one of five different kinds of actions  which are essentially characterized by what this set of limit points looks like.  So the first possibility is that our action is elliptic.  And H of E is the post-set which contains the elliptic structures.  And what this means is that there are no limit points of our group on the boundary of our  hyperbolic space.  And in terms of structures, it means that the elliptic structure, which is the equivalence  class of the generating set that contains every element of your group, is the smallest  structure.  And in fact, the associated telegraph is quasi-asymetric to a point.  The second possibility is that the action is linear.  And H sub L is where we collect all of our linear structures.  And what this means is that there are exactly two limit points on the boundary.  Moreover, it also means that our group contains a loxodromic element.  And by that, I mean that there is a bi-infinite axis in our space.  And this loxodromic element acts along this axis by translation.  And if I pick any other loxodromic any element in this setting, its limit points on the boundary  are the same as this other loxodromic element.  The third possibility is the one that I'm interested in the most, which is where we  collect quasi-parabolic structures.  And what's a quasi-parabolic action?  Well, it's where we have infinitely many limit points on the boundary of our hyperbolic space.  But moreover, we request that there is a global fixed point on the boundary.  And the last one is where we have our general type structures.  And in this case, we also have an infinite number of limit points on the boundary of  our hyperbolic space.  But the group has no fixed points on the boundary.  So the standard example to think of general type actions are free groups that act on the  heligraph that looks like an infinite tree, so for example, F2, which is the free group  on two generators, acting on the heligraph that looks like a four valence tree.  Now Gromo's work has one more possibility.  It contains the possibility that a group acting on a hyperbolic space can be parabolic.  However, parabolic actions are never co-bounded.  And because I'm dealing with actions on heligraphs, those actions do not turn up in this partially  ordered set.  And that's why I only have four substructures here and not five.  And why am I interested in H of G?  Well, it gives us a way of studying all possible co-bounded actions of our group simultaneously.  And the hope is that if I can understand each of these smaller partially ordered sets, then  there's a way to put that information together to try and understand H of G as a whole.  And so not only can I look at all the different actions of my group on hyperbolic spaces, there  might be from this examination a candidate for a best action to study in order to try  and understand my group via a group action.  And that brings me to HQ of P, which is the post set containing the quasi parabolic structures.  And the reason I'm particularly interested in them is because in our joint work with  Dennis and Carolyn, we could say a lot of things about linear structures and general  type structures.  But there were not a lot of things that we could say about HQP of G. And that's why  my interest lies in this post set.  Furthermore, it was clear to us that quasi parabolic structures do behave very differently  from some properties that we were able to determine for linear and general type structures.  And in order to illustrate that, I'm going to cite a few theorems from our work.  So it turns out that we can construct groups Gn, which have as many linear type structures  as we want, but they have absolutely no quasi parabolic or general type actions on hyperbolic  spaces.  And we can do the same for general type structures.  We can create groups Hn that have exactly n general type structures, but no quasi parabolic  or linear structures.  And the natural question to ask is, can we do the same for quasi parabolic structures?  And the answer is no.  Because if you have a quasi parabolic structure on your group, you most definitely will have  a linear structure on your group, and they will be related in the sense that the linear  structure is dominated by the quasi parabolic structure.  And the machinery to do this comes from using the Busaman pseudo character.  So a pseudo character is an almost homomorphism.  So it's a homomorphism up to some bounded defect.  And what we do is that we start with our quasi parabolic structure.  We look at the action, the quasi parabolic action of the group on this Cayley graph.  And when you run the Busaman pseudo character on it, what it does is create a projection  to a line.  And it does this by crushing enough of the action out in a manner that is analogous to  expanding your generating set.  And because of that, we get a linear structure.  And because we've expanded our generating set, and our dominance relation is inclusion  reversing, we get that the linear action is smaller than the quasi parabolic action.  Moreover, while we had lots of examples of groups that have different linear and general  type structures, we had very few examples of groups which admit quasi parabolic structures.  And one of the main ones in our paper was this group Z reads Z, where you can get an  anti chain of quasi parabolic structures on this group.  The way to do this is to factor through the action of ZN reads Z, which are the line fighter  groups acting on their bus their tree.  And if you order these by divisibility, then you can get an anti chain of quasi parabolic  structures here.  But we were unable to come up with examples of groups that have a finite number of quasi  parabolic structures.  So some open questions from our paper were the following.  Does there even exist a group that has a non zero finite number of quasi parabolic structures?  Is there a group where we can find an uncountable chain of quasi parabolic structures?  And if we request a more complicated example, is there a group where you can simultaneously  have a chain and anti chain of quasi parabolic structures?  And I'd like to mention that instead of quasi parabolic structures, if we ask the question  for general type structures, then the answer is yes.  In fact, you can pick your favorite a cylindrically hyperbolic group and HGT of that group will  always contain a chain and an anti chain of cardinality continuum.  So this makes it a natural question to ask in the other situation when you have an infinite  number of limit points on the boundary of your hyperbolic space.  And because we know that there is some relationship between linear and quasi parabolic actions,  it's natural to ask that if our group does have quasi parabolic actions, then is the  number of linear structures always less than or equal to the number of quasi parabolic structures?  In other words, is it always true that the linear action on such a group has to come  from applying the boseman pseudo character to some quasi parabolic structure?  And I was able to answer these questions subsequently after our joint work with Dennis and Carolyn  was done.  So here are the answers to these questions.  The lampliter groups, all of them, Zn reads Z for n at least two, have a finite number  of quasi parabolic structures.  There is a group, namely F2 reads product Z for which you can have an uncountable chain  and an uncountable anti chain of quasi parabolic structures.  And moreover, there exists a group where the number of linear structures strictly is larger  than the number of quasi parabolic structures, which is non zero.  And all three of these theorems are actually the consequence of a much more general theorem  that I proved for groups that look like G reads Z.  So here is that theorem.  We let G be a group.  And I claim that if I consider the hyperbolic structures of G reads Z, then I can always  find a substructure, which I call B of G sitting inside of it that embeds into it, in fact.  And what does B of G look like?  B of G has this structure.  So I consider the poset of proper subgroups of G ordered by inclusion, and I have these  two disjoint copies of S of G. And the reason they're disjoint is because every element  of this copy is going to be incomparable as a structure to every element of this copy  and vice versa.  However, every structure from these two copies of S of G dominates this common linear structure,  and that of course dominates the smallest structure, which is the action on a point.  And the two copies of SG correspond to quasi parabolic structures on our reads product.  Moreover, if G is a finite cyclic group of order n, then this embedding is an equality.  So for the lamplighter groups, this is exactly the structure of the hyperbolic structures  on our group.  And how do we prove this theorem?  Well, my work heavily relies on the work of Capras, Cornulia, Mono, and Tessera.  And they have this machinery of strictly confining automorphisms, which can be applied to groups  that have this semi direct structure.  So you have a semi direct product with Z.  And I'm not going to define what a strictly confining automorphism is because it's a bit  of a technical definition.  But what's important for the purpose of this talk is that if you start with a subset of  your base screw bay, that's strictly confining under either of the generators of Z, so T  or T inverse.  And from this information, we can extract a quasi parabolic structure on our group H.  And it is exactly the Kruglitz class of the generating set that contains this subset of  A along with T to the plus minus one.  And what does the quantifier regular mean?  Well, that means that if I take this quasi parabolic structure and I apply the Busaman  pseudo character to it, the linear structure that I get is actually the projection of this  group to the Z factor.  So the homomorphism that you end up with is the projection to Z.  And how does this apply to the group G read Z?  Well, we can write our read product as a semi direct product where the base group is the  direct sum of Z many copies of G. And how does T act on it?  Well, T will take this infinite tuple and shift every element one space to the right.  And T inverse will take the same tuple and shift every element one space to the left.  And now I pick my subgroup H of G. And I'm going to define sets which are subsets of  A that are going to be confining under the action of T and T inverse.  So I define Q H, which has this structure.  And I think of this as starting at index zero.  I am allowed to have elements of my full group.  But on all the negative indices, I only allow myself to put in elements of H. And Q prime  of H is going to be the mirror image.  So now starting from index zero all the way to the left, I have the full group.  But on the positive indices, I only allow elements of H. And the thing to notice that  this structure is compatible with the action of T because I can shift the full group.  I can shift the subgroup H into the full group, but I may not necessarily be able to shift  backwards.  And here this structure is compatible with the action of T inverse, but not T for exactly  the same reason.  And it turns out that Q H is actually strictly confining under T. And Q prime of H is strictly  confining under T inverse.  And when I do this process for every proper subgroup H of G, what I do is that I get an  entire collection of Q H's, all of which are confining under T. And I get another collection  Q prime of H, all of which are confining under T inverse.  And this is exactly what creates those two disjoint copies of S sub G in my partially  ordered set, one that corresponds to the Q H's and the other that corresponds to the  Q prime of H's.  And because of the incompatibility of the action of T inverse with this structure, and  similarly T with this structure, this is also intuitively what makes them incompatible  to each other in our post set.  And what does the regularity condition give us?  Well, it's important because it gives us the common linear structure.  Because no matter which one of these quasi-parable structures that I pick, and I apply the Busavan  pseudo character to it, I get exactly the same homomorphism to Z. So the linear structure  that they dominate from this construction is the same.  The hard part of the proof is showing that there is an equality when G is a finite cyclic  group of order N. And the reason is that you first have to argue that in this case, every  quasi-parabolic structure firstly comes from a confining subset.  And then you have to use long algebraic arguments to show that that confining subset has to  be the look like QH or Q prime of H for some proper subgroup H of G. But how does this  buy us the examples that we want?  Well, for the land-cliar groups, this is obvious because I have an equality.  And why do I have a finite number of quasi-parabolic structures?  Well, because Zn has only a finite number of subgroups.  And how many quasi-parabolic structures do I have?  It's twice the number of proper subgroups of Zn.  So that's a finite number.  And for the example, which is F2 reads Z, here's what we do.  We start with P of N. And this is the poset of the subsets of the natural numbers ordered  by inclusion.  And I embed this poset into the proper subgroups of F infinity.  So F infinity here is the free group on countably infinitely many generators.  And I start by indexing these generators.  And for any subset of the naturals, I map it to the subgroup that's generated by the  elements that have those specific indices.  And now you can embed F infinity into F2, which is the free group on two generators.  So you get an embedding to the proper subgroups of F2, and by the theorem, this creates an  embedding into the quasi-parabolic structures of F2 reads Z.  Now of course, I could have done this with if F infinity reads Z as well.  The reason I wanted F2 was because I wanted an example of a group that's finitely generated  with this property.  And lastly, what is the example of the group that has strictly more number of linear structures?  than quasi-parabolic structures?  I claim it's the group that looks like this.  It's the standard land fighter group direct product with Z.  This group only has two quasi-parabolic structures.  And those are precisely the quasi-parabolic structures that come from the land fighter  group.  The direct product with Z does not contribute any new quasi-parabolic structures.  However, I claim that this group has uncountably many linear structures.  And how do we get this?  Well, our group K has two independent projections to Z.  One comes from the Z factor of the wreath product, and the other comes from the projection  to this copy of Z.  So you can actually get a projection to Z cross Z.  And now you pick your favorite line in the Euclidean plane of whatever slope you want,  and I can project the entire grid down to this line.  And if I do this for lines of different slope, I get very different, in fact, incomparable  linear actions.  And because you have a continuum worth of slopes that you can choose, you get uncountably  many incomparable linear actions.  And that's why this is cardinality continuum.  So not only is the cardinality of the linear actions strictly greater, this is as different  as it can get.  So after I did this work corresponding to wreath products, Carolyn and Alex took this  machinery of confining subsets, and they applied it to other groups.  And the work of theirs that I want to mention here is the results that they have firstly  for bump-stalked solitaire groups BS1N.  These groups also have a semi-direct product structure, which looks like Z1N, semi-direct  product with Z.  And they were able to show that the hyperbolic structures of this group have this partially  ordered set structure.  So what's happening here?  Well, we take the prime decomposition of N, and we assume that there are K different  primes involved in the prime decomposition event, and this is that K.  And then you take the power set of the set 1 up to K. That creates an embedding of this  lattice here, and all of the structures here except the smallest one, which is the linear  action, correspond to quasi-powerballic structures.  In addition to these, there's one more that comes from the action on a hyperbolic plane.  And of course, then you have the linear structure dominating the action on a point.  But all of these quasi-powerballic structures also have interpretations in terms of arc and  finding subsets.  And the other result of theirs that I'd like to mention is the classification of hyperbolic  structures on the group Z squared, semi-direct product Z, where this extension is defined  by a matrix from SL2Z.  So it's a two by two matrix of determinant one.  And it turns out this is the structure we get.  So we have two quasi-powerballic structures, which dominate a common linear structure,  and that dominates the action on a point.  But in terms of confining subsets, what are these quasi-powerballic structures?  So we take this matrix phi and we consider it's attracting and repelling eigenline.  And in terms of confining subsets of Z squared, these are neighborhoods of the attracting  and the repelling eigenline for phi.  So together with Alex and Carolyn, we're currently working.  And what are some of the questions we're interested in?  Well, we want to know, does there exist a group that has an odd number of quasi-powerballic  structures?  Because all of the examples we know so far have an even number.  For the land-flatter groups, it's twice the number of proper subgroups of Zn.  That's an even number.  In the bombstock solitaire group case, it's also an even number.  And for Z squared, semi-direct product with Z, it's still an even number.  So does there exist a group that has an odd number of quasi-powerballic structures?  And in particular, does there exist a group that has only one?  Because what we'd like to do is construct groups KN that have as many quasi-powerballic  structures as we want.  And I'd like to mention that if I can answer this second question, then it would be immediately  possible to answer the third question by taking the n-fold direct product of this group.  Because for that group, the quasi-powerballic action would have to factor through one of  the copies of your group, and that is exactly what will give you exactly n.  And I'm also interested in understanding from the setting of the VEET product for which  groups the equality holds.  So I know it's true for the finite cyclic groups, but I also know it's not true in general.  And I also know that it's not true for every finite group.  There is a counter example in the case G is a Z2 direct product Z2.  So that's the finite cyclic group of order two direct product with itself.  But I suspect that this equality could hold at least for some of the finite symmetric groups.  So what conditions does your group G need to ensure that this equality holds?  And the work that we're focusing on is aimed at understanding the following.  We'd like to classify the structures on groups that look like Z to the n semi-direct product  with Z, where the extension is defined by SLNZ.  And what we suspect is that in this case, as long as you consider the eigenspaces of  a fee, your quasi-parabolic structures at least on this group should somehow be in some  one-to-one correspondence with those eigenspaces.  And what we'd like to do is also understand structures on iterated H&N extensions, with  the ultimate goal being to try and understand the structures of polycyclic groups.  But because these groups are not going to admit general type actions, the hope is that  maybe we can find specific examples that answer these questions in this class of groups without  having general type actions.  Because we know that probably we won't be able to control the number of linear structures.  But at least if we can have no general type actions and have as many quasi-parabolic structures  as we want, then we're closer to understanding H of G as a whole.  So that's sort of the direction in which we're headed.  And that's all I had to say.  Thank you very much.
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