 Okay, great, thanks, it's a pleasure to be here and good to see a lot of familiar faces. So I'm gonna talk about rationality in a sense completely different from the last talk. So let me start with some generalities on growth. And I'll start with a connection between the growth of finitely generated groups and with manifolds. Then I'll spend most of the talk on growth of groups, but in the end, give an application back to manifolds. That's kind of the plan. Okay, so, well, for a compact manifold, you can consider, okay, you can consider its fundamental group and as a finitely generated group, this carries various word metrics corresponding to choices of generators and you can let be the sequence of growth values. Also, if you wanna be careful, you should remember to decorate your notation with both the group and a choice of generating set. And by definition, these growth values just enumerate the number of group elements that you can spell with up to N letters from the generating alphabet. Well, if the manifold is compact, then this is related to the volume of the ball in the metric on the manifold, radius N. And they'll be related in the usual quasi-isometric way, which in particular preserves the fact of being polynomial if one of them is polynomial than the other one is and furthermore, the degree of polynomial growth. So each is bounded by a polynomial if the others. So for most of the talk, as I said, I wanna consider finitely generated groups and this discrete growth function. And then in the end, consider what that tells us about some corresponding volume growth questions in manifolds. Okay, so, oh, I should say, also when you change the generating set for a finitely generated group, you certainly change the sequence of numbers, but again, in a way, in the usual quasi-isometric equivalence so that not only is polynomial growth well-defined in its degree, also being bounded below by an exponential is UI invariant. Okay, so for example, non-elementary hyperbolic groups have exponential growth and they don't just have exponential growth in the sense of being bounded below by some exponential, but they have definite exponential growth in the sense that the growth function is bounded above and below by exponentials with the same base. This is a theorem of Cronert from the 1990s. Okay, so what are some groups with polynomial growth? Since the focus of this talk will be not only understanding when the growth function is bounded by a polynomial, but I wanna ask the more precise question when is the growth function precisely polynomial. To distinguish those, let me use the term polynomial growth range to emphasize the case of simply being bounded by a polynomial. Okay, so we've known, well, a simple example. Let's consider free abelian groups. So if you take the free abelian group Z squared with its standard generators, we get the familiar picture that looks like this. And in this case, we find that beta n is two n squared plus two n plus one. And as I said, here, we see that the sequence of growth values isn't just bounded by a polynomial. It actually records the values of a polynomial on the nose. And so I wanna regard that as a kind of nice arithmeticity and ask the question, when can you actually, within the polynomial growth range, when can you improve on this and get something stronger like this? That's the motivating question. Okay, so there's a classic result of Bath and Barsh from the early 70s, which says that no-potent groups are in the polynomial growth range. And just to fix ideas a little bit, let me write down their formula for the degree of polynomiality. So, well, what does it mean to be no-potent? It means that if you form the Law or Central series by successively defining gm plus one to the group generated by commutators of gn with g, then this eventually terminates in the trivial group. In other words, no-potent means that if you take commutators of commutators of commutators to some finite depth, everything in the group is killed. Well, here we have successive quotients that are abelian because you're quotienting by commutators and then some. And since they're abelian by the classification of abelian groups, they look like some as e to the dn cross finite. Okay, so you have this nice sequence of characteristic subgroups, which each of these grows like a polynomial of degree dn. What Bass and Givarsh tell us is that for these groups, you get polynomial growth of the order. This is sometimes called the homogeneous degree that you'd get by adding up, not the dn's themselves, but the dn's weighted by how deep in the Law or Central series that they are. In fact, I feel the situation was clarified further circa 2001. Olsen studied distortion of subgroups in no-potent groups, and what he showed is that if you look at each of these gn in the sequence, it's precisely power n distorted. All right, so that's a nice way to understand this formula. You're getting contributions from each of these z to the dn's that you see, and this factor is telling you how distorted it is in the ambient geometry of the no-potent group. Okay, so let me remind you a very brief description of the overall history of questions on rate of group growth. I'll recall, asked if all groups had to grow either in the polynomial range or be bounded by an exponential. Maybe there's a dichotomy between groups in those two ranges. In other words, is it possible for the betas to grow faster than any polynomial but slower than any exponential? And this was partly motivated by Nolner and Wolf proving that this dichotomy holds for solvable groups. Subsequently, we know that many families of groups have a teats alternative in which members of that family, such as, for instance, linear groups, linear groups are either virtually solvable or they contain free subgroups. And that means that for linear groups or in fact, any class of groups with a teats alternative, you'll never see growth in the intermediate range. As is well known, Gregorchuk found a counter example in the 1980s. There are groups of intermediate growth. And so it's possible, in fact, to see growth on the order of K to the square root of N. And interestingly, even though Gregorchuk and many others have worked on understanding groups of intermediate growth ever since then, they all grow with this rate. And so a modified Nolner conjecture is still alive. Maybe everything's in the polynomial range, exponential, or here. That would be sort of curious, but it's the best of our knowledge. That could still be true. Okay, and then, well, there's a kind of bombshell theorem of Gromov's from circa 1980, in which he does something very powerful. He gives a converse to Bass and Givarsh. So not only do nilpotent groups grow polynomially, but polynomial growth range implies virtually nilpotent. Okay, so that makes nilpotent groups, in terms of growth, a particularly interesting class. They're exactly the ones with polynomially bounded growth functions. Okay, so now what I wanna do is ask a kind of orthogonal question, as I said. So there's, when studying the growth of groups, you can ask how fast, and you can ask how regular. Let me explain what I mean by this, and then I'll fill in a little bit of information on this. Okay, so one very nice case, and that includes the class of exactly polynomial growth functions, is the case that the growth value satisfy a recursion. So in some cases, you'll have that the nth growth value is some linear combination of k previous ones. I like to think of this as a Fibonacci style recursion. Well, some recursion with finite depth. By the way, finite depth, I'd like to just make sure we're distinguishing that from, for instance, classically, say the partition function was studied by Euler and shown to satisfy recursions, but the depth of the recursion gets bigger for the larger partition values. So this is just the class where you actually have some finite depth. Okay, so, well, consider the actual Fibonacci numbers as an example. Take the Fibonacci numbers themselves. What happens is, have great success in studying them if you form the associated generating function. So the associated generating function is the formal power series, where you put in the values of the sequence as coefficients. And of course, the fact that you satisfy a recursion tells you something about this generating function when you shift it. And if you solve, in the Fibonacci case, you get this nice expression, one over one minus x minus x squared. Okay, so that persists. Any time you have a recursion, you can rephrase this in terms of some coincidences that happen with the generating function when, the generating function when you shift it by K different places and take linear combinations. So when this happens, the associated generating function is actually equal to a ratio of two polynomials. And the recursion even tells you, just like it did here, what coefficients you get in the denominator. So in this case, what we'll see is we'll get some sort of polynomial over one minus alpha n minus one x minus. Okay, so the polynomial that you have on the top is an artifact of what the initial values are here. In any case, you get here an actual rational function, rational in that sense, a ratio of two polynomials. With a little bit of thought, you can convince yourself that this is an f and only f. So satisfying a recursion of this kind, a rational recursion, is equivalent to being a rational function. Okay, so let me take a moment to say why this should be interesting if you wanna understand something about the group. So the idea is that, well, so if you're interested in studying finitely generated groups, there are a bunch of different hard decision problems in general that you can take on. One of them and maybe the most famous from Dain from a little over 100 years ago is to solve the word problem. Given a presentation for the group, decide whether a string of generators represents the identity. And another thing you might wanna do if you're a geometric group theorist is build the Cayley graph. You have some presentation and you wanna actually be able to draw what the Cayley graph looks like. So these are decision problems. And we know since the 1950s that this one is in general undecidable. There are groups for which there's no algorithm that will tell you whether a string of letters represents the identity. Well, so one thing to notice is that these problems are essentially equivalent. Well, deciding which words are, which strings of letters are trivial is the same thing as understanding where edges lead in the Cayley graph. A string of generators represents the identity if and only if the corresponding path leads back to the origin. And the thing that I wanna observe is that knowledge of the growth values, of course, if you could solve the word problem, you could figure out how many words there are of each length. And of course, if you know the Cayley graph, that tells you you just count points in the ball in the Cayley graph and that tells you the beta n. But the thing that I wanna point out is that the arrows go the other way as well. So if you know these numbers, if you know how many elements there are of a particular size that armed with that, you can solve the word problem becomes decidable. Briefly, that's because what's hard about the word problem is that if you search by sort of brute force, if you take a string and you start hitting it with relators and try by brute force to find whether there's a way to simplify it to the identity, you don't know when to stop. You don't know when to stop looking because you might have to make the word longer before you can make it shorter. So you can't tell the difference between I haven't found the simplification yet and there is no simplification. But if you know when to stop looking, if you know how many of the strings have to agree, which is what this is telling you, then that makes the problem decidable because it tells you that if you're still looking, you simply haven't found enough simplifications. Okay, so I claim that these growth functions encode all kinds of fundamental algorithmic information about the group and therefore, that's why we should care about rational growth because if your growth function has a nice recursion like this, that means that all the beta N values are determined by finitely many initial values. Yeah, oh yeah, that's right. So usually to get this equivalence, I wanna allow them to be in Q, although it's also an interesting case to consider the subclass where they're just in Z. But yeah, they're constants because I have a one here, so that's not the same thing. Idiot. Oh, I indexed these by the term that they're, yeah, I'm not sure about the question, I'll ask you later. Good, so this is why we care. What this is telling us, rationality, right, our priority, this is just a formal power series. And sometimes that power series happens to be a ratio of polynomials and what that's telling us is that we've got an algorithm with only finitely much information that does all kinds of decisions we might care about in the group. Okay, so what I wanna emphasize over here is that this question really is kind of orthogonal to the rate of growth. So we can have slow growth or fast growth, we can have rational or not, and there are examples in all quadrants. For example, the rational function one over one minus x has coefficients one, so that's polynomial in particular. On the other hand, one over one minus two x has coefficients two to the n, so you can have exponential growth in the rational class. Well, you can have, let me call this px over here, so a nice example, you can have the digits of pi power series, three plus one x plus four x squared plus one x cubed plus et cetera. A nice way to see that that's not rational as it was pointed out to me last week is to notice that if you plug in one tenth to this, you actually get pi, right? So if you had a rational expression for p of x, you plug in a tenth to that, and pi would be rational. And it isn't, okay, so here we have something slow growing in the sense that the coefficients are actually bounded, they're all between zero and nine. So we have something with globally bounded coefficients, but they're very erratic, they're transcendental in some fundamental sense. And so of course you can cook something up over here if I say, oh, my name. Okay, so it's in this sense, it's not the same question as how fast. It's asking kind of how erratically. Okay, so then there's a, the question of which groups have rational growth had a big, a lot of attention was given to it in the early 80s. And some combination of Canon, Chroma, and Thurston on one hand and Benson on the other showed that if G is hyperbolic, that's these guys, or virtually a billion, that's this guy, then beta n is rational. Yes, so let me say I'm going to adopt the terminology that if this generating function is a rational function, i.e. if the beta satisfy a recursion, I'm gonna call the sequence itself rational. I'll say that the group has rational or that presentation has rational growth. Okay, so is the same and clear? All right, so this is really strong because notice change of generating sets, as I said before, completely changes these betas. It changes them up to quasi-isometric equivalents, but that's worth nothing, right? Because after all the digits of pi, that's boundedly close to this one, so the constant sequence. But one is rational and the other is transcendental. So quasi-isometry is far too violent for the delicate condition of rationality. So it's very remarkable that you're able to say anything for all generating sets. Okay, so let me take a quick moment to sort of sketch in cartoon form how you might be able to do something like this. In brief, you wanna understand the geometry, right? And it's the geometry that's gonna give you a handle on how to count, right? So the point of this proliferation of names partly is that Canon gave a very beautiful thin triangles proof of this, but before the definition of hyperbolic groups existed. And then Grumov said, hey look, here's this other definition. Now your theorem holds for all hyperbolic groups and not just for manifold groups, which was the case that Canon was considering. Thurston motivated by these questions initiated the study of automatic groups. Okay, so let's see what a picture like this might look like. So let me just draw cartoon idea. About how something like this might go. So to do that, I wanna think about Z squared, the free billion group with its standard generators again. The same group that I was drawing before. And so let's notice that you get the following pretty nice picture. If you start at the origin, I'm gonna take the standard generators are, I'll call them A and B. A is the generator that goes that way and B goes this way. So if you start at the origin, there's a machine that can decide whether a string is geodesic. And that machine looks something like this. I can do a letter A as many times as I want. I can do a letter B as many times as I want. But if I follow a B by an A or an A by a B, then forever after I can just do positive A's and B's. And this will be geodesic. Any number of A's followed by any number of B's and A's, that's fine. What you can't do while being geodesic is ever have an A and an A inverse. You'd never do that in a billion group. So on the other hand, if I did label it like this, an A inverse that takes me over to this kind of other world that looks something like this. More colors would be better, but in the interest of time. Okay, so you get some picture that's something like this. And so this machine that you're thinking for you, and it encodes all the allowable moves. And as long as you follow the arrows of this machine, you're geodesic. Okay, and what's nice about a machine like this, a finite state automaton, if you will, is that it can therefore count geodesics because you can regard this as a directed graph, take its adjacency matrix, raise it to a power, and that counts the number of paths of each particular type. Okay, so does that ever see what I'm doing here? So once you have a machine like this with finitely many states that encodes geodesics, so this enumerates geodesics, that lets you count. And you can do something similar. So there's a problem, right? Because what we care about is group elements. And some group elements are reached by only one geodesic in this alphabet, like A to the 100, while others are reached by many geodesics. So if what we care about is enumerating group elements, this will over count. But instead you could come up with a related machine, and I don't want to make sure I'm doing this right. And you could think later on about how to decorate it. Okay, what you could do instead is come up with a related machine that doesn't over count. And the reason this one won't is because, yes, there are many geodesics to reach this point from the origin in the caligraph of Z squared. But if I restrict my attention to a subclass of geodesics, those that do all their A's first, followed by all their B's, that's a kind of normal form, right? And with these four different shapes, I can uniquely enumerate everything in the group. And so if you play around with this later, and I think this is fun, so maybe you should. You can collapse the machine a little bit into something that has you do all your A's first, followed by B's. So in the labeling over there, it would look something like this. And this collapsed machine actually enumerates group elements once a piece. You're a little bit careful. So okay, so this is the gateway to the theory, as I said, of automatic groups. And this idea works in the virtually a billion case. And something like it, Canon came up with something quite like it for the hyperbolic case. Just if you, the key words for the hyperbolic version of this are cone types and n types, if you get interested in looking this up. Canon's paper, which appeared in Geometry Dedicata and I think 83, is extremely beautiful and readable. But what he does is he exploits, as I said, thin triangles to sort of argue that locally, if you look at how geodesics can continue, there's kind of only finitely many local pictures. Okay, so what I hope is an obvious question is what about no potent groups? So we know they're in the polynomial growth range. Are they rational? Are they decided by finance data? Tom, Tom, can you take this theory and apply it directly? Right, I.e. the question is that. Okay, so a quick remark about this. If we're looking at no potent groups or virtually no potent groups, actually being polynomial, that might be your first question. As we had for z squared where the answer was two n squared plus two n plus one in standard generators. Well, in free a million groups, you always get polynomials on the nose. And so maybe your first dream would be that the growth functions are actually recording values of polynomials in no potent groups. Okay, so remark. Of course, it's easy to see that polynomial, so if the b and r actually polynomial, that would certainly imply a rational growth. There are lots of ways to see that, but one easy way to see that is that a polynomial satisfies a recursion with depth just given by its degree. Because you take successive differences enough times and you get zero. But actually this is too much to hope for. There's a slightly bigger class that's really the one that we have any hope of. So that slightly bigger class is called eventually quasi polynomial. And let me just make a very quick remark about what that is. Think about the example, for instance, of a crystallographic group. So suppose you have some group, like some wallpaper group that acts with some reflections on the plane. So that's virtually a billion, but there you might expect that the answers of the values for the growth function depend on parity. You get one kind of growth for even parity and the other kind of growth for odd parity. And indeed that's what happens. So there you're gonna get two different polynomials in the case of a reflection group in the plane. In general, two different polynomials, one for the even case and one for the odd case. Okay, and so that motivates the definition of eventually quasi polynomial. All that means is that there exists threshold and a period such that n equals, let's say a, okay, so eventually quasi polynomial means once you've surpassed this threshold time, the values are given by a polynomial, but it could be that these A i, which actually depends on n, unlike François in the earlier case. These are periodic. Right, so what I'm saying is, here I have the behavior that oscillates between agreeing with two different polynomials. Well, I could equally well interpret that as a single polynomial with coefficients that oscillate mod two in this case, does that make sense? And I'll just say a quick word. So it's very easy to see that these also satisfy recursions. You can make a modified argument for the polynomial one to see that these all have rational growth. And let me take a moment to say why this is what we should hope for. So our modified question then, this can't be true, but maybe in the best case what we'd find is that the growth functions of no-potent groups cycle between agreeing with finitely many polynomials. And that would be the best kind of arithmeticity we could get. Okay, so, well these guys, these EQPs, they come up everywhere. And one really classic place you might have run into them is in the so-called air heart theory from combinatorics or algebraic geometry. Air heart theory goes like this. If you take a polytope with rational vertices and you start to dilate it and you count how many lattice points are inside it, that enumeration sequence is eventually quasi polynomial. So it's a very natural class to come up in the context of counting functions. So again, I don't wanna write too much of this down but I'll just say these are kind of old friends if you're doing combinatorics of this kind, lattice combinatorics, because a big class of examples is dilates of polytopes and the number of lattice points inside them. So the number of lattice points inside the n times dilate of P is eventually quasi polynomial. So this is a classic fact from the 70s. Okay, so why are we seeing it here? Well, if you think about the example of z squared with its standard generators, these are the points you can reach with one step. These are the points you can reach with two, with three and so on. And the whole plane is exhausted by dilates of this original shape. And the number of group elements you can spell with a certain letters is exactly the number of lattice points that are inside a dilate of this original polygon here. Okay, so there's a close relationship, we shouldn't be surprised, between the group theoretic counting problem and the kind of combinatorial lattice enumeration problem for polytopes. Okay, so now let me say if the hope is that it's still not true. Okay, so bad news comes along in the 1990s in the form of Stoll's theorem. So Stoll tells us no, no potent groups need not have rational growth. And he gives a very beautiful, very simple example. And that example is the five parameter Heisenberg group. So that group lives in four by four matrices by putting five integer parameters in those positions. Five parameter Heisenberg group. This is a two step no potent group. If you apply the Baskie-Varsh formula, you find that the growth is polynomial of degree six, but that's just saying that it's bounded above and below by polynomials of degree six. Is it eventually quasi polynomial? Here's what Stoll tells us. Well, it can be rational or it can be transcendental. It's transcendental in the standard generators, which is sort of embarrassing. But it's rational in some other generating set as prime. So this also answers the question that I hope you were wondering, which is yes, a priori quasi isometry can destroy rationality, but does it ever actually do that for groups? Well, yes, it does. But to my knowledge, so this theorem comes from 1996 or the paper does anyway. To my knowledge, this is still the only example we know of a group that has rational growth in one generating set and not in another. I should say he does it for all the higher Heisenberg groups, five and up. I think that's still the only examples we have. Okay, let me say a quick word about how he does it because it's beautiful and it's also relevant for what I want to explain next. So what he does is, well, he shows that if, so suppose your growth values are on the order of alpha times n to the d. All right, that just means this ratio approaches one as n goes to infinity. So this is just the first order approximation. Then if alpha is irrational or transcendental as a number implies the same form. Okay, so let me repeat that because it's pretty cool. So if to first order, you have a coefficient that's not rational, then your associated generating function can't be a rational function. Not obvious, but the proof is like four lines long once you think of the right thing. So a quick example of this that's quite nice, I think, comes from the Gauss circle problem. Here we considered enumerating the number of lattice points in the dilate of a polytope. The Gauss circle problem asks how many z squared points are there in the disk? So here d is the unit disk. And if I dilate it to the disk of radius n, how many lattice points are inside that? It's called the Gauss circle problem because, well, the answer is roughly pi n squared because that's the area. But it's pi n squared plus some error term. And I should put a question mark here because this is actually still an open problem, but it's widely believed by a number of theorists that this is the optimal answer, that the error term looks like n to the one half plus epsilon for any epsilon bigger than zero. We know from Hardy and Littlewood that just playing one half doesn't suffice. Okay, so I just bring this up to say, Stoll tells us something about this. Stoll says, no matter how hard this, just because it has a first order term with a transcendental coefficient, no matter how hard it tries to satisfy a recursion, it can't. So Stoll's criterion tells us that this sequence of values here is a transcendental sequence. Okay, so that's how Stoll proves this part of the theorem. What he finds is a shape, a shape whose dilates enumerate the group elements. And then he takes the volume of that shape. Well, here's the picture. There's a whole theory by Ponsu and Successors that tells us that the asymptotic columns of a finitely generated nopotin groups are nopotin-lea groups with some funny metrics on them that I don't need to describe in great detail called CC metrics. They're sub-Rimani and are actually sub-Finsler metrics. But the point is what Ponsu tells us, put in kind of concrete terms, what Ponsu tells us is that if you take the ball of radius n and you pass to the limit, you stand really far away and you look at the shape of the ball of radius n. What Ponsu is telling you is that there's some sort of limit shape and at the end of this talk, I'll show you a picture or two of what these limit shapes actually look like. That's in some sort of associated lead group called the Molchev completion. Okay, from this theory, it follows something really nice, which is you get a very similar story to this one and to this one, namely, if you want to know about how big the growth is, you just take the volume of that shape. It's in this nice sub-Rimani in space. It has a volume form. Take the volume of that shape times and to the degree that was given to us by Bass and Kivarsh. Okay, and so how does Stoll prove this kind of remarkable theorem? He shows that if you take standard generators, the volume of that funny looking ball is five over 8748 minus one over 6561 log two. See that log two? That's not rational. So that's how this theorem works. And what he does on the other hand for the rational case is also beautiful, but in the interest of time, I won't say much about it. I'll just say he cooks up a generating set that works like this in the sense that the iterates completely exhaust the space with no gaps in between them, and then he shows rational volume for that modified generating set. Do I have any questions at this point? Okay, so no-plotin group that's even simpler than this. This is almost the simplest non-Abelian no-plotin group, but there's one I like even better. Which is the Heisenberg group itself. And Stoll's results leave open the possibility that the Heisenberg group is rational in all generators. Okay, so that turns out to be true. So this is joint work with Mike Shapiro. The hippie at Tufts, not the other Michael Shapiro at Michigan State that many of you probably know. I guess it's a common name. So the theorem is actually the Heisenberg group is pan-rational, i.e. rational in all generating sets. And what's kind of nice is that this is the only known example outside of hyperbolic and virtually Abelian. That's it. What also is kind of challenging about that, I mean people had thought about this before, but what's challenging about it is that none of the tools that work for cannon and crew apply here. It's not automatic. It doesn't have a regular language of geodesics. There are infinitely many cone types. There are infinitely many n types. And that whole laundry list of tools doesn't work. So instead you need to understand something pretty detailed about how geodesics look and then build a nice but much more complicated machine than the ones that I was drawing before. Questions about the statement? Yes, yeah, that's right. So let me emphasize that. So we have hyperbolic groups and virtually Abelian that are rational for all generators. There are some groups that are rational for no generators. For instance, intermediate growth groups and groups of unsolvable word problem, of course. Right? Those can never be rational for any generators. For the intermediate growth it's because if you think about the rational expression and you look at the poles, you look at the roots of the denominator, if those poles are inside the unit circle that makes the growth exponential. And like a 20 minute exercise will convince you that if all the poles are on or outside the unit circle that makes the growth polynomial. So intermediate growth can't possibly be rational. But there's also a middle column to have in your mind which is groups that might be rational with a special, especially nice generating set. So for instance, it's an exercise in Borba Ki that Coxeter groups have rational growth with standard generators. And there are lots of examples like that. There, some solvable groups are known to have, with nice generators, to have rational growth. I think a lot is known about Red and Goldarten groups. But the all generating sets move. That's been hard to show you the depth of our collective ignorance. I think that's not even known for F2 cross Z. So all generating sets, usually, this is why geometric group theorists avoid properties that depend on generators, right? Cause you don't have a lot of tools. All right, so let me say a bit about this. So one thing is, marks. So there are these limit shapes. Actually, I can, while I'm talking, I can pull up some pictures. I guess it'll take a minute, but eventually show up. So for the limit shapes, the volumes are rational. And that's actually not hard to see and I'll kind of show you in the pictures eventually, why this might be true. Yeah, okay. There's a lovely wooden model of the Cayley graph of the Heisenberg group and standard generators that Dylan Thurston made, packed and shredded newspaper and sent me in the mail. And so that's to illustrate, that's the standard generators, but I'm sorry, it's a little blurry. But the decision problem is pretty difficult. Like how to build out the Cayley graph from finitely much of its information? That's pretty hard. But in this case, it's rational. So this was proved actually in my collaborator, Mike Shapiro's thesis from the 80s. What he showed is that in standard generators, the growth function, this is the spherical growth, but that's almost the same as the ball growth, is actually, you should expect the spherical growth to grow again cubed because the ball growth grows again to the fourth. And he shows that it's actually this funny looking quasi polynomial where that constant term oscillates with period 12. And that's just the standard generators, okay? But what are these, what happens when you change generating sets? So here's a fairly mild and nonviolent change of generators where I take A, I take B, but I also take A times B as a third generator. So I get the points of this hexagon here as the projections to the AB plane of a generating set for the Heisenberg group. What are A and B, by the way? A is the elementary matrix in this position and B is the elementary matrix in this position and those suffice to generate the group. In joint work with Christopher Mooney, he and I study what the limit shape has to actually look like. The Ponsu limit shape, we want an actual procedure, an algorithmic procedure for building up the limit shape. Don't have time to tell you about it, but it amounts to solving some isoparametric problems in the plane, there's only finitely many ways to do that and bang, you get some picture that looks like this. Okay, so we have an algorithmic process that starts with generators and it builds this shape. I've drawn it in three dimensions because there are three parameters here so I get to do that. And that shape has flat walls and this funny curved roof which you get by graphing finitely many different polynomials and piecing them together. Okay, so these are the wacky looking limit shapes for the Heisenberg group and their volumes are rational because our process for building them up just builds them as graphs of quadratic polynomials and you want the volume you just integrate some polynomials with rational coefficients to get a rational value. I'll leave that up for now. Okay, so the volumes are rational, that's actually, so this algorithm, this joint work with Christopher Mooney from I think about 2008, so we've known that for a while but I just want to observe this is harder because Stoll's criterion doesn't go both ways, right? A rational leading coefficient is far from enough to guarantee rational growth overall. Okay, so I just have two minutes left. Maybe I'll just say, I'll literally wave my hands a little bit about some of the ideas that go into this. The point is that this third coordinate, the C coordinate, the central direction in the Heisenberg group, the most efficient way to travel very far, it's quadratically distorted per OSIN and the most efficient way to travel in that direction is to enclose a lot of area in the AB plane. In other words, A to the N, B to the N, A to the minus N, B to the minus N, some great big commutator carries you up to a big value in the corner position when you do the matrix multiplication. And that means that that's why isoparametric problems are coming up, to move around efficiently you want to enclose a lot of area. Okay, and so essentially the way that Mike and I approach this is to really understand shapes that the geodesics want to take and then we build a machine that says to represent each group element geodesically there are really only finitely many viable candidates and they can be compared linearly. So you end up with this kind of linear comparison process over finitely many candidates and rationality ensues in very brief. Okay, and I wanted to add that quick remark here. Actually, maybe the 30 years later perspective on Gromov's theorem is that this limit shaped business is not incidental, it's fundamental to what's going on with polynomial growth. In a way what Gromov's proof amounts to is the only way to get polynomial growth is to have a shape that you dilate and count points in. If you unpack Gromov's theorem that's kind of actually how the logic goes. And no-putting groups are the only ones with dilations. That's actually a characterization of no-putting groups as it turns out. Okay, so in the last one minute I wanted to come full circle and give an application back to manifolds. So without writing anything down I'll just say this. This Heisenberg stuff is gonna tell us something about manifolds. Namely, it's gonna tell us something about cusp complex hyperbolic surfaces. So the reason is that if you're the quotient of complex hyperbolic space CH2 by a discrete group then your universal cover is the complex hyperbolic space CH2 and in that space the horospheres carry no-putting geometry. In particular, if you're the fundamental group of a CH2 manifold complex hyperbolic surface you're relatively hyperbolic and your parabolic are this guy. So there's a theorem from the 90s of Neumann and Shapiro in which they show that on that list of groups that are rational with good generating sets you have relatively hyperbolic groups with virtually a billion parabolic so in other words the real hyperbolic case but they're only able to show that in that paper for special generating sets. Nice ones that corresponds to the automatic structure. So in work in progress with Mike Shapiro what we are doing right now we think we can lift that for relatively hyperbolic groups. So we think that if you're relatively hyperbolic rel abelian or Heisenberg we have an argument to show that you're pan-rational that for any generating set you should always get a kind of arithmetically growing counting function. Okay, good place to stop, so thank you.