 So thank you and I would like to thank the organizers for organizing this summer school. It's great to see so many young people interested in group theory here in Bure. I will speak actually about isoprometry and growth of groups. We'll be considering a finitely generated group G. S will denote a finite generating set. We'll usually assume that S is symmetric. And I recall that the growth function counts the number of elements of length at most L in the word metric associated to S. Another word counts the size of the balls in the calligraphs in the calligraph of G. It's clear that the growth function is a sub-multiplicative because to write the word of length n plus m there's at most V of n possibilities for the first n letters at most V of m possibilities for the next n letters. And in particular the growth function of n plus 1 is not greater than the continuity of S V of n. Since the function is sub-multiplicative if we take log of the growth function this is a sub-editive function. So if we divide it by n as it's the case for any sub-editive function there is a limit. It turns to some limit. Or equivalently we can take n's power of V of n and there exists a limit as n tends to infinity and we denote this limit by the letter V. It's also clear the growth function is sub-multiplicative so clearly the growth can't be more than exponential. And so we say that if V is greater than 1 then there is a low bound which is exponential. We say in this case that G is of exponential growth. For example as you probably all know if you take a free group on let's say D generators with D at least 2 then this so the calligraph is a regular tree. So then the cardinality of the sphere first of all we have at most 2D possibilities for the first letter V of n plus 1 minus V of n. So is it most clearly 2D and then for each next letter we have at most 2D minus 1 possibilities so it's clear that a free group is of exponential growth. And since it's very easy to see if you have a free subgroup then the group also has exponential growth because subgroup clearly gives an estimate for the growth of the group. In particular I forgot to mention so it's clear that if we change the genetic set the growth function can change but it's a synthetics clearly doesn't change. If you have a free subgroup the growth is at least exponential and there are really many many very important classes of groups which do admit free subgroups like for example tits alternative says that any linear group unless it's virtually solvable has a free subgroup and actually it's enough even to have a free sub-semin group clearly. Sorry Mark if you have some group G that admits a free sub-semin group then the growth is exponential. No just I prefer V notation for just first I said that there exists limit if you take log divided by n and or equivalently if you take nth root of V of n then there exists a limit so sometimes indeed sometimes one denotes by V the log rather log of our V but I prefer to denote this V the nth root so if you have a free sub-semin group clearly these trees inside the group give you a lower bound which is exponential so in particular it is well known that by result of Milner and Volfe if the group is solvable either it is virtually important in this case the growth is polynomial or it has a free sub-semin group then the growth is clearly exponential and once more revising tits alternative so unless the group is virtually important the growth is exponential so we see so when I say the growth is polynomial so if V of n is smaller than n to some power then we say the growth is polynomial and by polynomial growth theorem of Gromov which is quite a deep result so it can happen even only if the group is virtually important and in this case we can say more not only that they have an upper bound but really up to a multiplicative constant V of n should be n to the d but then we can say if V of n is neither polynomial or exponential then we say the growth is intermediate then and it's not so easy to construct some groups where the growth is intermediate and today we want to have some first glance on the still mysterious and rich class of groups of intermediate growth and first I would like to remind a conjecture the gap conjecture which is due to Grigorchuk that says that if V of n is not polynomial V of n is great let's say greater x some constant square root of n or if we change the constant the same is to say for all sufficiently large n there is a weaker form on this conjecture that asks for x n to some alpha some some positive alpha and even this weaker form is very very open so some motivation for this conjecture formulated by Grigorchuk yes first of all groups of intermediate growth were found by Grigorchuk in the 80s and this kind of estimate holds for all his examples and there are some general results if you for some classes of groups if g is rigidly rigidly important then the result of Grigorchuk also in the 80s as this is true much more recent result of John Wilson if if g is rigidly solvable then at least a weak form of the conjecture is true with V of n greater x some constant n to the power one over six unless the growth is polynomial there are some more general results in this direction but very very little is known unless we assume some additional properties like here on the group so the only really very general result is due to you who the Shalom and Therese Stau who show the following that g for any finitely generated group g if V of n is not polynomial then V of n is greater than n to the power log log n to some power where c is some small absolute constant that the growth are the polynomial or greater than constant square root of n yes certainly both are possible for example if g is really important for example if g is a billion it is clearly reasonably important so it could be polynomial but true in the sense the conjecture is to other polynomial or greater than x square root of n so now these general results the only really general result we have this result is n to the log log n to some power so you see it's much weaker than what is asked so instead of having instead of having x n to some alpha what is the weak form of Grigorychuk's conjecture ask we have x well constant log n log log n so instead of n to the alpha we have some log n log log n so far before we discuss the groups of intermediate growth I would like to recall some other definitions and we will apart from the growth function there is one other function we would like to consider namely the Ferner function so first of all I recall that a finally generated g is a finite group g is amenable if the following holds if for any epsilon greater than zero there exists a finite subset V in G such that the size of its boundary is epsilon times smaller than its size and here the size here the boundary means the following when we speak about the boundary it's a discrete analog discrete notion of the boundary so we look on the calligraph we have some subset V for example here and the boundary is defined as points of V a distance one from the complement of V such that the distance a in the belt metric associated to S from V to G minus V the set V is equal to one so for example here in this picture this square lattice was supposed to be a calligraph of Z2 and here's our set V this point is not the point of the boundary it's point of V but the distance one we don't have points outside of V but all these points for them are points of the boundary because here at distance one you have always have something not in V here I gave the definition for for finally generated groups the definition can be adapted for countable groups but for us it's sufficient to know that first of all this definition for finally generated groups it's not difficult to check it doesn't change us if we change the generating set S and then if you take a countable group a countable group is a minimal if and only if any finally generated subgroup is amenable so we can use it as a general definition for for countable groups and there are many many examples of groups which are which are amenable there are many many examples of groups which are not amenable sometimes it could be difficult even for a given group to understand whether it's amenable or not so on Wednesday we'll have lectures on Thompson groups where it's a big challenge to understand whether it's amenable or not and but we need one more definition where is the razor thanks we'll need the definition of the feulner function given g and s the feulner function measures the best possible size of the sets with small boundaries so the feulner function of n is defined as minimal of the cardinality of V where the minimum is taken over finite sets V such that the boundary of V divided by the size of the boundary V divided by the size of V is smaller or equal than 1 divided by n so we see the feulner function is well defined if and only if the group is amenable so evaluating the feulner function means solving there's a permanent problem in our group so when we want to understand what is the feulner function of some group first of all we want some upper bound and though it could be tricky in some complicated groups they don't even understand whether they amenable not in but in many groups it's not so difficult to find some examples of the feulner sets the sense like this with the small boundary are called feulner sets but the main problem to understand the asymptotic of the feulner function is to find the lower bound of this function meaning to prove some asymmetric inequality in our group so in the exercise list you will have the example that in the free group you will ask to ask you to find best possible constant for linear is a permanent inequality in the free group so free group can be can be shown to be non amenable and unless you know it already so you will have it in an exercise and one one can find best possible so in the free group example the boundary is always greater than the cardinality of the set times some constant and so the main problem about feulner function is the amenable case we would like to understand the asymptotics of this function and again when in some groups we do understand it in some groups it could be also a difficult problem to to find the lower bound now before we formulate some general open questions so we would like to prove now the following result due to Coulomb and Salaf Kost that says that the feulner function is always asymptotically larger than the growth function when I say asymptotically larger if one function is asymptotically larger than the other one if f2 of n is smaller some constant f2 some constant here of n it's for any group yeah this is for any group let's see let's see that though it's not important let's say so it makes j is any finitely generated group yes it's true also for this is exactly the command let's assume that j is infinite though inequality makes sense for finite groups as well so when I say greater equal in fact I don't want to write in now but there are explicit constants for this inequality that we will see right now in the proof so which makes some sense even for finite groups but our main interest are infinite groups indeed is there is some way to to get the middle one I would like to thanks a lot thanks Kate so let us take some set v so v is finite is a finite subset in our group g let us assume that its boundary is small with respect to its size so let us assume dv the cardinality of dv divided by cardinality of v is not greater than one over n and we want to show that v cannot be too small we want to show that the size of v is up to multiplicative constant in the argument not smaller than the growth function of n so first we want to choose some ball which is of more or less the same size as v namely let's look on the minimal possible n such that the growth function of n that is the size of the ball of radius n is greater or equal than two times the cardinality of v since we assume that our group is infinite it's clear that v of n tends to infinity as n tends to infinity larger and larger balls obviously cover all our group so whatever number we fix here we can always find some some some ball which is larger and we take minimal n with this property then since n is minimal observe that v g s of n is not greater than two cardinality of s cardinality of v because we had a remark that v of n plus one is not greater than cardinality of s multiplied by v of n so if if we would have something larger here if not right then not only for n this is true this would be true for n minus one as well so we can we have n with these two properties now we'll do the following thing we will consider all possible points v in v and we will draw a ball of radius n centered at v so we will consider the balls centered at v of radius n for all v in v excuse me yes at least at least the same n in the series of the boundary this is this the same n yes that's the right to one so this was the argument so we have fixed really this n minimal possible n with this property and then we observe that using sub multiplicativity using the fact that v of n plus one as we stated already is not greater than cardinality of s times v of n right this was just a particular case of sub multiplicativity so that I have mentioned already right if for this n for this particular n if you would have here something larger than this so if v of n would be larger than two s cardinality of v then using this we would claim that v of n minus one is was also great or equal to cardinality of v and this is in contradiction with what we assumed we took a minimal possible n you define a n before the n is defined by the boundary of v so that's why that's ah yes sorry sorry yes it's not it's not the same n thanks a lot so this is m let's say so now now we continue to prove so in any point v of v we draw a ball of this radius m and we want to count total number of points which outside of our set v taking an account the multiplicity so we let's n the number of points in all these balls outside of v with multiplicities now the word so we take any point which is in some of these balls and outside our v we count in how many balls this point we have this point and we take the sum of all over all points out such v of this number of points and this is our total number of n outside v and we want to find two possible ways to estimate to estimate n what we can say on one hand for any ball on one hand n is greater than cardinality of v because we have so many charges just for the center of our balls and in each v so we have at least cardinality of our ball so v gs of m minus cardinality of v we have at least so many points outside of our set v because at most at most cardinality of v could be inside so all the rest is outside and by our assumption this is greater than cardinality of v and this is greater than cardinality of v so this is greater cardinality of v squared on the other hand we want to to get an upper bound for our number capital n we do the following thing we consider our ball centered that identity for the moment of the same radius as all these balls we are considering here this is the ball we want to take the points of this ball and join them with jadezik by jadeziks with the center of the ball so we add and add jadezik like this so we get some spanning tree so we fix some spanning tree in in the ball of the radius we consider that joins points paths are jadeziks between some points x and identity between so we fix some spanning tree like this and we consider shifts of the same spanning tree for all vertices v so we can consider then this was spanning tree so let's denote it by comma and we want to shift it so have it in all points so for all v we consider v times gamma so then we will have the same basically the same tree everywhere for our balls now take some in some ball take some point which is outside so we have some point let's say y here look on the this jadezik path inside gamma between y and v we look on that path between y and v and look what happens y is outside our set v this small v is inside our set v meaning and and this is a jadezik path then clearly this path intersects at least once the boundary of v so here this there is a boundary we we take some x consider if there are several let's say closes to y to y x on the boundary of v and on on this path x on the path between y and v now observe even if you would not wouldn't know our center we can fix the following information so again we have some ball here there is a center v here somewhere the boundary of our set v here we had some point y in some jadezik path intersecting the boundary suppose we would know v minus one times times y meaning we know this quotient we know what is the role of y inside our ball and suppose we would know also the distance we know suppose we know we know y v minus y v minus v minus one times y to know this element and if you wouldn't know and suppose we know this distance distance distance between y and x suppose we know this element only and this distance then i want to say that we we we can figure out where is where is the center of the ball if we know this distance if i yes and we know yes i we know i forgot to say we know x rather we know x we know this point and we know this distance and we know the this element then we can figure out where the center was meaning that if we we can say that our cardinality n then is at most the following thing so we have at most v of m possibilities of for this element v minus one y just possibilities what is this element in the ball we have at most at most m elements for m possible choices for the distance since this is ball of radius m so this distance is not not more than m times times times m and we have all possible choices on the boundary of v and so we will supply by the size of our boundary v here there are only two three so what have we we got this we can make it a bit on one hand n is smaller what we have written here so on one hand n is smaller than v of m and v of m we knew that v of m is at most so this this was at most two cardinality of s which is just some constant cardinality of v and then we have m and cardinality of dv so on one hand our capital n is smaller some constants which is two times cardinality of s cardinality of v multiplied by m and cardinality of dv and on the other hand n was larger than cardinality of v squared and so we received what we want so we just divide by cardinality of v we obtained that dv cardinality of dv divided by cardinality of v is great or equal then what we take dv is some constant which is one divided by two times cardinality of s sub constant divided by m so if the size of v was approximately v of m this was our assumption at the beginning of the proof then we concluded that the boundary should be greater than this so we have proved what we need to prove in colon salve coste inequality let's have a look on some examples and let's take a look when this inequality is asymptotically sharp first i'm genuinely marked what kind of examples of amenable groups we know so if you have zd for example as the two actually we have drawn already the picture if we take some zd if we take some cube and of the size and to the d it's boundary so this is v is a cube it's boundary is so the cardinality of v is equal to this and the boundary is some constant times approximately some constants times and to the d minus one for example on the squares here the cardinality of the square is n square and the cardinality is of the perimeter is something like 4n or 4n plus one right so very easy to see in a billion groups there are sets simply cubes like this with the boundary much smaller than these sets and instead of considering the cubes like this we could consider simply the balls alternative we could consider the balls of radius n in our billion group and then also easy to see we have the same thing so the cardinality of the ball is n to the d and the size of the boundary that is the size of the sphere vn is is approximately n to d minus one more generally if if we simply know that the growth bound more generally another example if you would assume that v of n is smaller than some constant n to the d it's clear that there exists a subsequence such that vn plus one minus v of n is smaller some constant n to d minus one if such sequence doesn't exist we would multiply then if you consider vn plus one divided by v of n we would multiply each times by one plus one over n and we would get a contradiction so just elementary observation just using the growth if we see that the balls are nice certain sets at least for a subsequence it's a little bit less it's less straightforward to use only the growth but after all since we know that the group of polynomial growth are virtually important the same true not only for subsequence the same is actually true for all n so the balls are asymptotically optimal for an asset so because we know that general lower bound is colon cell of cost and just looking on the observation that the spheres are much smaller than the balls in important groups we see that colon cell of cost of inequality is asymptotically optimal for all groups of polynomial growth now what happens for groups of exponential growth for groups of exponential growth for some groups the inequality is still optimal and for some others it could be very far from being optimal but let's have a look on some optimal example I recall that a reef product of two groups and b is an extension of the sum of copies of b and index by a so we take just a direct sum of b's and a acts on this direct sum just by shifts on the index set we take the extension this is called a reef product I prefer an old-style Ukrainian notation for me it's group a that acts so there are two possible notations for reef products sometimes this group is denoted by a reef product b and sometimes b reef product of a one should take care reading the papers but anyway this is an extension of two groups like this this is called reef product so some very basic and well-known examples is something that called lamplight a group group is when our group that a could be for example z but what is important our group b has two elements is b is a finite group consisting of two elements the corresponding product so a with b is called lamplight a group why because it's easy to visualize the elements of this reef product and also it's easy to visualize the biometric for this reef product why this group is called lamplight a we can imagine the following thing we have z or some other base group for us set for a moment and suppose that in each point of z we have some lamp what happens with our word matrix so we have two generators one generator is generator of z so elements of our groups will be pairs some z and z and f by definition here f is some element here if we can interpret this f as a function from from z to z to z and it's a finitely supported function how so how we view this element so we say z well z is some some element in z and this function we look on its support if some elements in the support we say the lamp is on in this point and if it's not in the support we say the lamp is off what happens with the word metric so there are two gene two generators in our reef product one is generator of z if you multiply it we just take a step on z without changing configuration at all in the other generator is is the lamp generator meaning we don't work on z we just change the lamp where we stand all the other lamps are left as they will so this is the lamplighter group so let's see that colonel of costa inequality is asymptotically sharp for this lamplighter group why say clone salaf costa bound is sharp is asymptotically sharp for for z reef product z to z or if there's any other finite group consider so we we fix some n consider an interval let's say from minus n to n in z this is some optimal formula set in z right and so we say z is in is this integral minus nn and say that the support of f is also belongs for the same interval then the cardinality of what is the cardinality of this set we consider so the cardinality is our set v is two to the power two n plus one because there are so many possibilities for lamps we we consider two n plus one points in each time could be on and off gives us add two to the power two n plus one and there are two n plus one possibilities for our point z so we have some sets growing exponentially and one can observe that cardinality of v divided by cardinality of the boundary divided by the cardinality of v is not more than one over n why just consider all possible choices of our configuration having this choice we have two n plus one possibilities let's say it's two i don't know is it one that is two divided by two n plus one so for each possible for each fix configuration we have two n plus one possibilities for our point z and only for the end points of the corresponding interval the point will be on the boundary of our set so among for each fixed generators there are two points among two n plus one which are on the boundary so we see so we have this we see the further function is at most exponential by this easy example and by Coulomb's self constant inequality we know it couldn't be smaller than exponential so it is optimal there are many other groups where the inequality is optical for example for any politically groups for those who know this definition it is optimal or for groups of finite proof rank for example but there are many examples and no difficult examples where the inequality is far from being optimal we will discuss maybe in the moment what are known results in this direction but now i would like to formulate one more conjecture which is more recent so gap conjecture for growth is one very one of the famous open questions in geometric group theory and there is a more recent conjecture this time due to also due to Grigorychuk but together with Pansyuk which says the following for any let's say infinite finally generated group G generated G either the further function is polynomial or the further function is at least exponential greater than x some constant and if this conjecture would be true it shows that for intermediate growth groups Coulomb's self-cost is never optimal we will in in the next lecture today we will discuss the first example of groups intermediate growth the first Grigorychuk group and for this group the growth is approximately x and to some alpha this alpha smaller than one and so Coulomb's self-cost to just give us this lower bound x and to the alpha but conjecturally it can be always improved in this conjecture obviously also very far from being from being solved and in remaining how many minutes i have three minutes just some more general words about as a primitive of in the intermediate growth case yes general mark so in the in the gap conjecture that I just erased for the growth functions there is this lower bound x and to the alpha and it's not clear at all what are best possible alpha to conjecture well maybe one half if you believe in a strong form but it's it's not a priori it's not clear at all what is really the minimal possible bound what is interesting in the gap conjecture for for as a primitive though it's a bit less known maybe at enter general gap conjecture for growth that if true it's clearly optimal this easy examples which example that we have just seen there are groups of exponential growth there the finite portion is exponential so you can't certainly can't ask nothing larger than exponential and the conjecture ask this is always the case just some obvious remark that I have not done that if growth is sub-exponential of is sub-exponential then the group is immutable growth g exponential then g is immutable again so I explained before for for polynomial growth case but again there is the proof is that subsequence for subsequence sequence of possible radii and i and the boundary of the ball of radius n i should be should if you divide by cardinality of the ball this should go to zero because if not if there is some positive constant as a low bound then we would get that vf for all n or for all sufficiently large and let's say v of n plus one would be greater that vn times one plus c for all sufficiently large n then we would get that v of n is greater one plus c n minus some constant so and exponential so it's clear if the growth is sub-exponential some some some some bolts are further sets but in contrast with the polynomial growth case we discussed there is no reason to believe that the the furnace sets we get are optimal right so in polynomial growth case they were really n times smaller the the the size of the boundary of the ball of radius n was n times smaller here the only I think we can claim in general the the ratio should go to zero ah yes sorry there is boundaries so amazing the ratio of the boundary thank you of the ball could not be constant times the cardinality of the ball because otherwise each time we would multiply the growth function by some constant greater than one maybe I have so two minutes left right to to give quickly to start giving the definition of of the Grigoryuk group that we will discuss in the next lecture because if you look on the exercise list all exercise but the last one you know already all the definitions so you can try to solve them there is one marked with star on one hand because it it is more complicated than the other exercises and on the other hand that I was not sure I have time to to to give the definition of the Grigoryuk group so maybe it's an exercise rather for not for this problem session but one exercise left for the future for you but I just very very briefly sketch the definition of the first Grigoryuk group but I will really explain it after after the break in the second lecture others on here thank you we'll consider a rooted binary tree like this you will consider four automorphism of this rooted boundary tree we'll have a which interchanges the large branches of this tree and doesn't do anything else this will be our generator a and then there will be generators b cd and their action on this bar on this tree would would be defined recursively so for example this element b we say first on the first level it doesn't move big branches the first branch will go to the first branch and the second will go to the second and on this first branch actually it will x like a this notation we use means that we identify this sub tree with a large tree and a meaning interchanging the the branches of this next level and on this on this point we'll say it acts like c what does it mean x i c c we have not designed yet but we say if we identify this branch with the whole tree the action of b should be the same as the action of c and then for the c we write a similar thing c does not change also the big branches but x as a interchanging the the branches of next level on the first branch in x like d here and d is slightly different because it's extremely on the first branch so also the branches are preserved extremely on the first branch and x as b on the second branch so i have to finish this definition here this definition if you have not seen it takes some time to to understand good right so i will try to explain you in more detail in the next lecture but at least i started formally i gave you the definition of the first gregerchuk group that that you see in the last exercise in the exercise list