 First of all, I wanted to say what a pleasure and an honour it is to speak at this conference for Alan. So, happy birthday Alan. Okay, so I just want to start off with just some motivation for what I'm going to talk about. I imagine that everybody here knows what a Caeligraff is since you're all friends of Alan. I don't see why he'd be friends with someone who doesn't know what a Caeligraff is. But I'll just remind you, just in case. So, we have a group G, which is finely generated by a set S. So, for the rest of this talk, the groups will be finely generated. The Caeligraff, so the vertex set of the Caeligraff is just the group G itself. And then recall that we connect to elements of the group, if we can jump from one to the other using one of the elements in our finite generating set. Okay, so this gives us a graph. And the properties of this graph mean that we really can investigate the group using this geometric object. So, okay, so this is a graph, so it's a metric space. Just using the shortest path metric. The group acts by isometries on this graph by left multiplication. I see we can use the group action as well. Okay, and so there are many connections between properties of this graph, so geometric properties of this graph, and algebraic or structural properties of the group. So, most famously, Gromov's polynomial growth theorem, which says that a group is virtually no potent, if and only if the Caeligraff has polynomial growth. Okay, so that is if we fix some element and we look at balls about that element, if the number of elements in such a ball grows polynomially with the radius, then we say that the group has polynomial growth. Okay, so that's all very nice. Oh yeah, and I should mention also, obviously if we change the generating set S, then we get a very similar object, we get something that's quasi-isometric, so something that is essentially the same as the Caeligraff with the other generating set up to finite perturbations of the metric. So, okay, and the object I'll talk about now is a kind of spin-off of this idea, so if our group has lots and lots of finite quotients, so from now on our groups will be residual finite. Okay, so recall residual finite means that for any non-trivial element in our group, we can find some finite quotients of our group in which that element remains non-trivial. Okay, so if we restrict to such groups, it would be a shame not to use the fact that we know that it has so many finite quotients in order to investigate the group, so what we can do is we can actually use the Caeligraffs of the finite quotients to make a space which kind of encodes the geometric behaviour of all of these finite quotients. Okay, so that's what's known as a box space, and so to make a box space what we do is we take some sequence, so we'll take a sequence of subgroups of G, and we'll call it a filtration if each of these is a normal subgroup of G. Sometimes we want it to be nested, so we'll maybe have that for now. It's not always necessary, but most importantly we want these to be a finite index as well. These are finite index normal subgroups, and we want them to have trivial intersection, so that when we take successive quotients by normal subgroups in the sequence, we will in a way approximate the group. We're going to get closer and closer to G. The object that we can make using such a filtration is a box space. G is residually finite, finely generated. The box space of G with respect to a filtration, and I, which we'll write in this way, is the following object. As a set, it's just the district union of these finite quotients of G by the various subgroups and I, and we want it to be a metric space. It's going to play the role of a Cayley graph for us, so we want to put a metric on it. I'm going to put the following metric, D, which there's a very natural thing to do on each finite piece in this district union. If we fix some generating set S of G, then we can induce a Cayley graph metric on each of these finite objects just via the image of S under the quotient map. D restricted to each of these pieces is just the Cayley graph metric induced by S. That's all we care about, because we want to study the geometric behaviour of all these finite quotients uniformly. A nice way to do that is to put them all into one metric space, so I'm going to define a distance between each of the pieces, but don't worry too much about that, because what's important really is that you have a space, just one space which contains all of these finite quotients. I'm just going to set this distance to be... I make some choice of this distance so that this distance is bigger than the maximum of the diameters of these finite quotients. This choice doesn't matter very much. I just want to be able to talk about all of the finite quotients together without overlapping each other. That's a box space. It turns out that this is also quite a nice object which can detect lots of properties of the group G. First of all, what are some of the properties of this object which allow us to detect these various properties? First of all, just as in the case of the Cayley graph, if we change the generating set, we get an object which is equivalent in some sense. It's invariant in a certain sense under change of the generating set, and what is the correct notion of invariance here? This is something called coarse equivalence. If we have two metric spaces x and y, then f is a coarse equivalence. If there exist some maps which bound how much f distorts the distances when we go from x to y. We have some control maps, row minus and row plus, which just go from the positive reels to the positive reels. We want these to be increasing, and we want the limit of these maps to tend to infinity. We want these maps to bound how much the metric is distorted when we pass from x to y. We want the distance in y of the image of some point x to the image of some point y to be bounded by these functions of the original distances. Not necessarily, no. This is a little bit like a quasi-asymmetric embedding at the moment. If you just replace these row plus and row minus by linear maps, then you get something stronger, a quasi-asymmetric embedding. Here we don't care about these maps too much as long as they tend to infinity. Obviously, that would be a course embedding, but we want a course equivalent. We also want some constant, such that the image of x is coosly denser in y. Such that, let's just say, from y and y, the distance between the image of x to y is bounded by c. That's the correct notion of equivalence for this talk. We have this nice invariance under the change of the generating set in particular. What I meant is that if I take the box space with respect to some metric induced by a generating set, then I'll write this for course equivalence. We'll have that these two things are coosly equivalence. I'll change my generating set to some s prime. Another nice property of these spaces is that we can recover the information that's given to us by the calligraph if we're willing to go far enough along our filtration. If we choose some radius, then if we choose an index big enough, the r-balls in the box space from that point onwards will look exactly like r-balls in the group, in the calligraph of the group. For all r, there exists some i, such that for all n bigger than or equal to i, the ball in the calligraph of g of radius r is just going to be isometric to the ball in the box space, the ball in any of the pieces of the box space from the point i onwards of radius r. We can see any finite piece of our calligraph that we wish from some point onwards in our box space. Okay, good. As I mentioned, there's lots of nice connections between geometric properties of this box space and properties of the group. Just to give you a motivating example, if we take g with property t, cash dance property t, so this is a rigidity property of actions on Hilbert space, then if we look at the pieces of the box space, then these pieces will be expanders with a uniform expansion constant. What is an expander? I'm going to talk more about this. I'll just define it quickly for you. An expander sequence is a sequence of finite graphs which are extremely well connected, but yet do not have too many edges. We want the following properties to hold. We want these graphs to get bigger and bigger. We want the degree to be bounded by some uniform constant for all of the vertices in all of the xi. We want the chiga constant, which is a measure of connectivity to be bounded from below for each of these graphs. I've messed up my board technique. She taught me how to use three boards because I came from the UK, a country where we can only afford two boards. She taught me how to use the three boards. She told me to always start in the middle and then go to the front and then go to the back. I'm sorry, Ella. What can I do? Let's do it like this. What's the chiga constant? I'll just quickly define it for you. If we have some finite graphics, we look at some finite subset A and X. We look at the boundary of A, which is the number of edges that we move in order to disconnect A from the rest of the graph. For example, this is my X and here's A. It's all of these edges here which are connecting A to the rest of the graph. That's the boundary. I need to divide through by the number of vertices in A and I take the infimum over this, and I guess I should take A to B at most half the number of vertices of X. This is the chiga constant. It's measuring how easy it is to disconnect X, but it's taking into account the size of the subset that we're actually disconnecting. It is a measure of connectivity. We have the definition of expander stuff. This is a result of Margulis, I should say. It was him who first realised that if you take a property T group, then you get an expander as the box base. This was actually the first explicit construction of expander graphs. These are very useful objects in computer science and network design and this gives you a systematic way to create these graphs. I think that's enough motivation for why these are nice objects. What I'd like to talk about is some questions concerning the rigidity of these objects. As for motivation, I just want to mention the following purely algebraic question, which I think is not directly related, but I think it's an algebraic version of the question that we're going to answer in a second. Imagine I give you a residual finite group and I'm going to look at the following object associated to this group. I'm going to look at the set of isomorphism classes of finite quotients of this group. It's going to be the set of finite quotients of G up to isomorphism. I give you the set of finite quotients up to isomorphism of this residual finite group and I ask you, can you tell me what the group is? How rigid is the group under this algebraic object here? In particular, a question of Remeslinikov, which is still unanswered, is as follows. Can you actually tell that suppose I give you the set and you can see it's actually the set of all finite quotients of a free group? If I give you such a set for a residual finite group and I tell you it's exactly the same set of... Anna, what is that letter? I guess it's a C. But it can be whatever you want it to be. C for class, I guess. Yeah, you're right. Why is it a C? I'm not sure. Yeah, I should have made it an A or something in honour of Allah. OK, so, yeah, if I tell you that the set of finite quotients of a group is exactly the same as that of a free group, can I actually say that my group is free? OK, and this is still unknown, this is still open, although there has been some progress on this by Brydson, Conda and Reid. If you assume some additional things about your group G, if you assume that it belongs in a certain geometric class, then you can conclude that free groups are rigid in this way, but in general it's open, which is kind of weird. G is finally generated, yeah. So, is this the equivalent sign for isomorphism? Yes, yes, this is isomorphism. Can you conclude that G really is a free group? OK, and so, with this question in mind, let's think about what this could be in the world of geometry. So, the question we're going to think about is, suppose that I have some resusally finite group G and I have a filtration, I take the box space and then I take another group H, suppose I know that the box space is geometrically similar, what can I conclude about the groups G and H? OK, so that's going to be what we'll be looking at. So, in the definition of these box spaces, you took the G over Ni with the world metric, but you made them disjoin with some fixed distances which are large enough and they were arbitrary. So, this is a choice. So, the choice actually doesn't make a difference for the course equivalence class of the box spaces. So, if I choose the distances to be growing as I did because I wanted them to be bigger than the diameters of the pieces that I'm separating, then actually this choice doesn't make any difference with respect to the course equivalence class. So, OK. So, the first result that we have in this direction is joint with Ella. So, what we were able to observe is that, if you have this course equivalence between two box spaces, then this actually implies a geometric similarity between the groups. So, here I mean the calligraphs of G and H. So, it actually implies that G and H are quasi-asymetric. And this probably isn't very surprising because as I told you, as we move along the sequence that makes up the box space, we see bigger and bigger pieces of the calligraph and it turns out that you can actually glue these things together to form a quasi-asymetry between the whole calligraphy. OK. What can we say about the opposite direction? It's not true. In fact, for the opposite direction, it's even kind of difficult to see what kind of question you might ask because something I forgot to mention is that, so if you just fix your group G but you vary the filtration, you actually get wildly differing course properties. So, you can actually have groups for which, with respect to some filtration, you actually get expanders and with respect to another filtration, you get something that embeds into Hilbert space which is kind of an opposite behaviour. So, you can really get all kinds of behaviour if you vary the filtration enough. So, it's not really clear what would be the... If there is economic filtration, you can just take all. Oh, you can take all of them. Yeah, you could. In fact, so, Thibaut Della B, who's here somewhere, he's investigated these objects as well. This is something we call full box spaces. This is also an interesting question, what happens when you take all of them. But here, in general, we take them to be nested, so we don't take all of them. It's also interesting. So, in your statement, G has property, T implies that all of these bones... Yes, exactly. There, it's all of them. For the free group, you have mixed situation. Okay, so this is kind of... Maybe not that surprising, as I was saying, but you can get some nice corollaries from this. So, in particular, one can answer the following question of Mandela Naur, who asked... Remember, we had these expanders, which are very nice course objects, which people are interested in. But how many do we actually construct, for example, via these groups of property? So, how many expanders are there, whereby this, of course, I mean up to course equivalents? Okay, so perhaps there's only really one course equivalents class of expanders that would be kind of a shame. And indeed, this result allows us to answer this question. Well, actually, sorry, I should say, of course, that Mandela Naur kind of answered their own question in a sense, so they constructed two examples of expanders which were closely different, where one had a large girth, where girth is the length of the smallest cycle, so they constructed a sequence where the girth tensed infinity, and another sequence where the girth does not, and proved that these were not coarsely equivalent. But there was no kind of scheme to show that any of the other examples are coarsely different. So, using this result with Alan, we're able to show that, for example, if you take some groups which you know are not coarsely asymmetric, so, for example, if you take various groups of property T, you can actually get infinitely many coarsely equivalence classes of expanders from property T groups. You can get continuum many, but I guess for continuum many, you kind of have to cheat a little bit, you have to vary the sequence of subgroups that you take. So, yeah, you can get continuum many. So, there are no continuum many groups of property T or two possible? Oh, I don't know, I just want some. Yes, sorry. Oh, that's a different question. Well, is it? Well, it's a really good question. Oh, there are? Yeah, then you can just play with it. Oh, then yeah, okay, then yes. Yeah, think of this as continuum in that case. So, for example, if you take various SLN, so, by a result of Eskin, if you take SLNZ and SLMZ for N not equal to N, then these are not coarsely asymmetric groups, and so the expanders that you get from these groups will not be coarsely equivalent. Okay, so maybe I should also mention some related results. So, David Hume also proved that there is a continuum of non-coarsely equivalent expanders, but using different methods which only work for expanders of large girth, so with girth tending to infinity, he used something called separation profile, which was invented by Benjamini Shraman Timar for different purposes, I think. So, okay, so, yeah, that gives you a whole other bunch of expanders, and I should also mention a result by Cajaldas, who proved that if two-box space is a coarsely equivalent, then this actually implies a uniform measure equivalence for the groups. So, he kind of extended this result in a measure-theoretic direction. Okay, and maybe another thing I should mention is a result by Delat and Vigolo, who proved that, who did a similar kind of thing for warped cones, which are a kind of sister object of box spaces, which allow you to construct interesting examples of finer graphs using group actions, rather than quotients. Okay, so now, let's go on to something different. Okay, so, we know from calligraphs that it's very satisfying when we can assume some geometric property for the calligraph and have some kind of algebraic or analytic or structural conclusion for the group. And so, it would be nice if, with these box spaces, if we could assume some kind of geometric information about the box spaces, so if we could assume that the box spaces are coarsely equivalent, but get some kind of structural information about the groups, okay? Here, we could get some geometric information, but what about some kind of algebraic information? So, this brings me on to the next part of the talk, and this is joint work with Thibaut de Labis, who is presumably still in the same place as he was before. And Thibaut is our son with Alain, so he's a joint PhD student. And we're very proud of him. Okay, so, yeah, just some motivation for this then. So, okay, so what we'd like to do is we'd like to take this geometric equivalence between the box spaces and deduce something about, for example, the subgroups which we actually used to create these spaces. So, okay, imagine if our group was free, okay? If I look at the fundamental group, that's a really good way of detecting, so if I look at the fundamental group of a caligraph, of a quotient, of a free group by some normal subgroup, then exactly what I'll get is the subgroup N, okay? So, for the case of free groups, I have this very nice way of detecting the subgroup by which I've quotiented by looking at the fundamental group, okay? Remember, the fundamental group is the loops that you have in your space, and here, of course, we've made exactly the elements of N into loops, so we detect exactly this group, okay? In general, this may not work, okay? Well, in fact, it won't work if I look at the caligraph of some non-free group quotiented by some subgroup, okay? This will not be equal to N, okay? And why is that? Well, if I think about the loops that I've got in this space, I'm not just detecting the elements that are in N, I'm also detecting the elements which are already loops in G, okay? So, the... If she's pi1, that's a caligraph of the free group quotiented by some normal subgroup. So, with respect to the free generating set of Fn. But I'm confused. If pi1 for graph, isn't that always a free group? Yeah, exactly. But N is a free group. Everything's fine. In the end, everything's fine. Yeah, so, yeah, exactly. Exactly. So, yeah, the pi1 of a graph is always a free group, so that's a reason in some sense why this doesn't work for a general group G, right? Because, you know, if you think about what this quotient will look like, your... as loops in the space, okay, you've got the elements that are now trivial, the elements of N, but you also had already loops in the caligraph of G, so you're also going to be detecting those. So, that's no good. So, what you'd like to do is you'd like to be able to mimic this behaviour in some sense for a non-free group, okay? And the way to do that is via these course fundamental groups. So, we need a course version of pi1, and, in fact, there already existed one, which was created by Barsalo, Cremor, Laubenbacher and Weaver, okay? Okay, so, let me talk you through what this course version is. So, we're going to define what our paths are, first of all. So, we're going to fix some parameter R for example, R equals 60. I really tried to get 60 into my talk, but there's nothing, there's no numbers. So, yeah, if you fix some kind of parameter R, we're going to look at the fundamental group on this scale, okay? So, what I'm going to define, first of all, are our paths. Okay, and our paths are just going to be maps from a discrete set of integers up to, say, the length of this path, let's call this map P, that's going to be a path. So, in the normal fundamental group case, paths are continuous maps from the interval to your space. Here, we've got this map, which is going from a discrete set of cardinality length of this path to our space X, so, yeah, let's fix some metric space X. For us, it's just going to be a caligraph, so, let's fix some base points as well. And we want this map to be our Lipschitz. Okay, that's going to be our replacement of continuity in this discrete case. So, and of course, now we want to define what the correct notion of homotopy will be for these discrete paths. And in order to define that, I'm just going to, first of all, define another notion called our closeness. So, I'm going to say that two paths P and Q are going to be our close if one of the following two conditions holds. So, one of them is quite sort of obvious. Okay, if I have some path, okay, maybe this is my R path in my graph X. Okay, I'm going to say that this other path is R close to it if it has the same length so if L of P and L of Q are equal and each of these distances is bounded by R. Okay, so that's a very natural notion of equivalence between these paths. And the other version just takes it into account what happens when P and Q have different lengths. So, I'm going to also allow them to be R close if so one path does something. Okay, maybe the other path is longer than it, so what it does is it just does exactly the same thing as this path but then it just stays here. Okay, so it just waits for the other path at the end. Okay, so this doesn't involve R but R is involved there. Okay, and now I can define my notion of our homotopy. This is just called the end or the middle or? Just at the end, yeah. It's just allowed to wait for it at the end. Is this what I would call fellow traffic? Ah, yeah, possibly. Yeah, I guess it is, yeah. Does fellow travelling have a parameter associated to it? I guess if you compose these two moves then it's going to end up being fellow travelling actually probably, yeah. So what we're going to do now is we're going to be able to use both of these moves to move from one path to another path. Except I guess what we're going to lose is the control on the parameter at each, oh yeah, I'm just going to write it down then. So P and Q are our homotopic. If there is some sequence of paths Pi such that we start off at P we move up to some path Pn, which is going to be Q and each pair of paths, each pair of consecutive paths are going to be our close. So we can make these moves via these two relations of our closeness from P to Q. But I guess what we do is, we don't care about how big N gets. So we might be making a lot of these moves so that we kind of lose control on this parameter R. We can no longer say that the paths P and Q are very close but we can get from one to the other by some number of paths which are close to each other. And the endpoints are the same or they can change also, thank you. Yeah, the endpoints can change because here they just have to be close. Okay, so that's a discrete version of homotopy and now we can define this discrete fundamental group for coarse fundamental group in the obvious way. So this coarse fundamental group at scale R of X with base point E is just going to be the set of loops based at E, so well let's say R loops based at E up to R homotopy. Okay, and so why is this a good idea? What can we do with this? Well, the main thing is that now using these moves of our closeness, we can actually, via our homotopy, jump over holes of size around 2R or R if you want. So now if we have some path, if we have some loop that does something like this, if this length is less than 2R or R, something in terms of R, then this we can actually homotop down to just the trivial loop in our space. Okay, so we can ignore short loops. And what this allows us to do is the following thing. So now if we take a group in which we can actually control how big, remember the problem here, why didn't pi1 work is because in this pi1 we got the loops of G mixed up among the loops from N. So if the loops coming from N and the loops coming from G are far enough apart in length, then we can actually pick a parameter R, take the fundamental group at scale R and it's going to kill off all the groups that are coming from G and it's going to preserve all the ones that are coming from N. So that's the idea here. So what kind of group can we do that in? Well, it's a finely presented group, right? So if we have G a finely presented group and I'm going to call some things something, or maybe not, maybe. So maybe I'll just take some normal subgroup N and G. Now if my parameter R is sufficiently far between the longest loop in G, so the longest length of a relator in G but it's smaller than the shortest non-trivial element in N. Yeah, the length coming from the word length on G. Yeah. Yes, oh yes, oh sorry, I see what you mean. Yeah, and here obviously it's the length in the free group on the generating set of G. So if I take this parameter sufficiently far between these two things, then actually if I look at, if I look at pi 1 at the scale of my caligraph of this quotient based at the identity, then it will actually detect the normal subgroup by which I've quotiented. Okay. So. Is N simply a finite index or? No, no. So the notion of homotopy is that the two loops are homotopic if they are homotopic through loops based on SE because otherwise like in topology you can have free homotopy of loops which gives something else. Yeah, yeah, so yeah we base them at base at E, yeah. I guess this is kind of the same thing as if you, you know, you take your caligraph of G modern and then you just glue in some cells of, for cells of size less than 2R. So you kind of kill off the, the relators that are coming from G in this way. Okay, and so why is, what is the, a good context in which we can apply this? Well, box spaces are a very good context. So now we can answer our question in a more structural algebraic way. So now if we assume that G is finally presented and residual finite, then if we have some course equivalence between two box spaces, well this in particular also implies that H is finally presented because the groups will be quasi-asymetric and so what this results on the course fundamental group gives is simply, it's simply telling us that actually there's an isomorphism between pretty much all of these subgroups. So I'm going to write this like this. So this is isomorphism, whenever I write two lines here, it's an isomorphism. So I write this in quotation marks just because there could be some small perturbations for finally many of these guys where maybe they're not isomorphic but in the long run they will just be isomorphic and maybe up to some finite permutations. Essentially you're getting this. So in particular this implies that the groups are commensurable which is a very strong structural conclusion. So commensurable just means that they share isomorphic finite index subgroups. But yeah of course saying this is stronger. So just some comments about the proof of this. So it's not absolutely immediate but this actually uses the result with Alain. So in the result with Alain what ends up happening is that each of the pieces that are making up the box spaces you actually get a uniform quasi-osometry between these pieces and then you can use that in order to apply this to the course fundamental group. So and how would you apply that? Well actually what you end up getting is that if you have some let's say calligraph of some group A and you have a quasi-osometry of that with some calligraph of a group B then what you end up getting is that there's a map, there's a subjective map from the course fundamental group at scale r of A onto the course fundamental group of B except it's not with the same parameter you actually have to change the parameter a little bit according to the quasi-osometry constant C. So whatever quasi-osometry constant C you have here controlling your quasi-osometry you actually get a subjective map from this course fundamental group onto a course fundamental group of your other group but at a slight cost you have to change the parameter slightly. But what you can do in box spaces because you have this limiting behaviour that the NI are getting smaller and smaller it's a filtration, they have trivial intersection so what you can do is actually you can go back the other way okay and so you can go back the other way via a suggestion of the cost again of changing the parameter here but if your NI is sufficiently far along in your sequence then actually this will just stay the same. So what you end up getting is a suggestion of this group onto itself it's isomorphic to the subgroup by which we quotiented in the case of our box spaces and because we're taking residually finite groups these groups are actually hopfian. So it actually uses hopficity here in quite a funny way so we're actually using the hopficity of the subgroups in the filtration. So okay great I'm almost out of time so just some small remarks about this result and some consequences so of course just as in the result with Alain you can also conclude that some objects which are constructed in this way are coarsely different just as we did with expanders showing that there are infinitely many cost-equivalence class of expanders so you can do such things for example you can prove that there's infinitely many cost-equivalence classes containing Ramanujan graphs I mean being Ramanujan isn't a course invariant but you have infinitely many cost-equivalence classes containing Ramanujan graphs but a more interesting kind of strange consequence of this is that you can have the following situation because of the very strong rigidity of these objects so we're used to with Cayley graphs if we have groups which are the same up to finite index we're used to the Cayley graphs just being quasi-asymetric that's something we've all kind of internalised in the case of book space you can have some weird situations so I have the following corollary so there exists a group G with filtrations Mi and Ni such that there is some uniform bound let's say D such that the index of Ni is always bounded by D in Mi so these subgroups are very, very close to each other in a uniform way all the time but such that the box spaces with respect to these subgroups are actually different in the same group G nested sorry, yeah, thank you indeed, yeah so we always have this and the index of Ni is always bounded in Mi and yet these box spaces that we're getting are not going to be costly equivalent so if you think about it this condition means that the G mod Ni are just bounded covers of these G mod Mi with this bounded index here but yet these objects have very different course behaviour which is kind of strange and just to finish I just wanted to mention another result by Vigolo so Vigolo exploited this kind of course fundamental group framework in order to prove that actually for these warped cones so these objects which are created using group actions on manifolds there actually are some examples of expanders which are coarsely simply connected so which have trivial course fundamental group and that proves that they're completely different from any of these expanders constructed using this box space structure so any of these expanders coming from property T groups or others so they're these warped space expanders with trivial course fundamental group really a very new example of expanders that we have okay, I'll finish there, thank you so if you apply the last theorem you mentioned to try to understand the remeslinic of questions so why you can't solve this question I guess the remeslinic of question is more about the kind of the algebraic nature of the quotions so a lot of these results in that direction they use things like the profinant completion which is like an algebraic way of encoding these quotions whereas we assume something a priori stronger well something different something geometric about these quotions in fact you see these box spaces are much finer objects than profinant completions because if you just to give an example so if you take a profinant completion with respect to some sequence and with respect to an interlicking nested sequence of subgroups then the profinant completion you end up getting is the same and a lot of these results which explore this object use profinant completions whereas box spaces you can have these kind of interlinking subgroups but which give you again box spaces with very different course properties so box spaces are finer objects and thus assuming some kind of equivalence between them it's a stronger statement so we're assuming something stronger so unfortunately it can't seem to shed any light on this kind of thing so if you choose the subgroup to produce box spaces which are equivalent then you deduce that the two groups are commensurable and what is missing is that you want an isomorph is there something that you didn't expect? When you assume that the box spaces are closely equivalent that's a strong geometric assumption whereas here what you would be assuming is that really the set of isomorphism classes of all the finite quotions is just the same but you don't know anything about how the quotions will behave geometrically so if you fix just because you know what the set of finite quotions might be you don't know that there will be the same set but it won't be the same set geometrically possibly so you need to fix generators for H when they behave in the same way as the generators Exactly, and you can't do that Yeah, it's to do with choosing the generating set for those that it works Is the theorem okay when G is just profanically profanically presented? Ah So what does it mean profanically profanically presented? Let's say if profanically profanically presented is profanically presented as profanically presented possibly it might work I'm not 100% sure as it stands but it might also work I know it doesn't work for finally generated groups in general you can have some examples very simple examples using Wreath products for which the groups are the groups are not commensurable but the box spaces are even just isometrically box spaces you can get with finally generated groups but yeah I would have to check, that's a nice question