# Random walks and random group extensions (GGD/GEAR Seminar)

boundary because different boundaries can arise so we’ll see more specifically and also does it have positive speed so if I if I take ten ten steps how far am I and so you can define speed we’re going to see it in a second so fit same simple examples okay example zero will be the following you take for instance of z2 and which acts for instance on two on itself so you have you think of it as a grid like this you start to say from a point zero and I have to give you what measure well one measure would be you just can go in all four directions with the same probability oops ok so you have 1/4 Delta mass let’s say 1 0 Delta massive minus 1 0 plus Delta a sub 0 1 plus Delta M assets 0 minus 1 so now I’m sure most of you have seen this example before so some of you can tell me what happens if you run this random walk from some point that’s this is escape to infinity now in this case clearly it doesn’t so in this case the random walk is recurrent so it comes back to 0 infinitely often with probability 1 of course in this case why well because in this case well make sure if we make sure that you have chosen the probability measure which is which is centered meaning that the probability you know the barycentre of this of this Delta Mascis is 0 so you have that here you have that basically the right the expect right so if you take so the expected value of of the increment is 0 so if you take if you define the end the distance between W and X and X and you look at the end over N well in this case this tends to 0 almost surely so the speed is 0 my speed we mean we can define the speed to be for instance the layman yeah limit as n goes to infinity of the distance with you w ant X X okay so I’m sure probably all of us in this example before right okay so so first this is the case that here the case is the group is a million and the geometry is flat somehow so we don’t like this I mean this is not the kind of groups that you are like we want to consider we want to consider groups where the geometry is actually negatively curved somehow as the group is not going to be a billion so the other example is when you take the free group on two generators which acts on itself and then you take you take again a centered distribution like for instance one quarter you take a with probability one quarter or even worse or P or B inverse so here what does the picture look like if you take the Kayla graph of F – everybody knows that so please answer yeah is this 4-valent tree and here okay here does a bizarre thing which happens so of course you can go four ways with the same probability so in some sense the distribution is still centered however what happens in this case it’s very different

we actually claim that here the speed or the drift sometimes this is called is actually positive and you’re actually and the random walk almost surely converges to the boundary of the space so what is the boundary of the space well is this Cantor set of all geodesics you should think about it is it has this nice topology of a Cantor set and so so why is this true so it’s quite simple actually even though the random walk does not have a preferred direction to go it still goes to infinity and that’s strictly related to the geometry of this object which is a tree so here ax has negative curvature in the sense that for instance is a hyperbolic in the sense of drama so so why why does that random walk converge to infinity so so here’s the trick again I consider the end this the sequence of distances from the origin and I ask you what is the average expected value of the end does anybody have a clue Oh so quite 3/4 but of course you’re on the right track so what happens so let’s think about you see what’s what can be if you take the n plus 1 minus the N so are you going further from the origin or backwards well you can go in further with probability 3/4 and backwards with probability 1/4 so what’s the answer so every time is 3/4 minus 1/4 so the expected value would be 1/2 and so here it’s about and over – when I say about because there’s one exception when you’re at the origin you cannot go backwards however that doesn’t happen so often anyways here you know of course everything can be worked out pretty explicitly but this gives us a feel of what’s going on so it turns out that the speed is actually 1/2 and so as I said even though we don’t know where the random walk is going we know that almost surely is leaving in compact sets and since this is actually a tree well if it lives compact sets and just come back well it has to converge in the boundary ok so that’s sort of the sort of gives you the geometric intuition of what’s going on of course you’re going to generalize this example very much and actually the proof was not no longer be related at all with with this case but anyways it’s important to keep this case any questions okay so let’s put down some more technical definition so okay we start with X we have okay separable technical reasons realm of hyperbolic space matrix place so of course glom of hyperbolic means that triangles are thin so they look like this and it looks like a tree so of course this doesn’t have apply to this case where the geometry is well and which we take we have a group again which acts by Osama trees and we take a measurement on G and of course we don’t want to do something completely stupid with this measure so I don’t know we didn’t want to take the identity over and over again otherwise you’re not gonna anywhere so we define Muir to be elementary non elementary if the support of MU so that’s the subset of the group then you

take you take a sub semi group generated by the supportive mule contains two independent loxodrome elements okay so by the way since we’re going to need it later we remember that the translation length of an element just defined as the limit of the distance G an axe axe and so log should Roenick means that the translation distance is positive and what does it mean that they are independent independent means that there are just the lots of romics have two fixed points on the boundary each of them g8 1 and independent means that the fixed points that’s formed two to disjoin topples so you have somehow you have three group inside you could you could prove that but anyways yeah we’re gonna need this none element RIT condition okay so once you have that we can run around the walk and ask for the questions that I said before so note there’s one thing that we do not assume that X is a proper space proper means that the clothes balls are compact or well similar question would be locally compact it’s not quite equivalent but similar condition we don’t assume either of those and it may sound bizarre but maybe not now because in topology now we have a lot of spaces which are hyperbolic and they are not proper and we can of course mention a few of them later but so this is a central assumption that’s really the main key of their novelty of of this work because more or less when X was proper the theory has been worked out many years ago actually so here’s the main theorems okay as I said everything is like here so you have separable space you have hyperbola city you have this non elementary measure so first so the is that for almost every right so yeah almost every simple path it converges to the boundary so there is a boundary point where X is the gram of boundary such that this actually converges to the world so the second one is that basically you have positive drift so you can say that there exists a positive else which that for instance if you take the limit of the distances you divide by n let me if for in this generality the limit may may even be infinite but what we done we can prove it it’s not zero so basically it’s larger than of course well sometimes we are interested in a more tame measure so if if mu has a finite first moment what does it mean it means that the average step is bounded so of course it’s not quite so you can have your measure your measure can have infinite support so if the measure has finite support is like really throwing dice they have finally many possibilities this time you could still have infinitely many possibilities but this condition tells you that okay you can go kind of far with you can you can produce long long steps but on average the distance you travel is mounted so of course this also puts an upper bound so you as also finest first moment then you really have the limits so the limit exists right in this case the limit

would be finite yeah now let me say a few more consequences so where’s other consequence are well for instance there if you look at the translation lengths of the generic element this is also is going to be not just positive but you will be growing linearly so the probability that the translation length of WN is a smaller than L times n goes to 0 as an just thinking in particular the translation length is positive almost surely but it’s stronger than that it grows linearly essentially then there’s very other things you can say well one other thing but I really just say it briefly if the action is a cylindrical plus some consequences on the measure so for instance the first logarithmic moment and the various other things and finite entropy these are the standard conditions then the gram of boundary is also the Poisson boundary of the random walk in the model for the Poisson the pissant banger is something which is attached economically to every random walk and doesn’t have anything to do with the geometry but then there’s all this crap all this people who ask well can we find a geometric object which represent this but to say it precisely one has to talk about harmonic function so I will not say it now but but anyways this is also a condition but only at least if it’s a cylindrical but the rest doesn’t doesn’t use the s Olinda colleague the other thing the final thing is the application to random group extensions or random in general random groups yes well you can never tell right if the if the person boundary is is everything or or the measure doesn’t charge something I mean somehow funked orally they are they serve the same purpose so you could say yeah you could say the Poisson boundary is a set of minimal illumination so you could say is the set of unique legardie combinations of course there’s always a measure attached to this so depending on some measure there’s some measure on the boundary for which you view you satisfy all the properties of the Poisson model so it’s all match it’s all in the measurable category so yeah other questions yeah I will now let me just finish say say the last the last part so the random group xn so how do we find how do you define a random group so there’s various models so one simple simple model is the following so you fix K and stop some integer and you consider K independent random walks so you do this process k times w1 absolute wnk independent random walks and then you consider the group beyond an end generated by this independent random walks and now the theorem says that almost surely yeah basically well we could say yeah with probability which goes to one so of the group then with write the probability that this group gamma and quasi asymmetrically embeds in X tends to 1 right so question symmetric we embeds means you have a

metric on the group which is the word metric and then you have a metric on the space and you want to ask yeah the map is the ornament yes so the map is right if you have WN I you map to WN I okay so okay so and then we’ll get to see a few a few more color is but okay so whose theorem is this so basically most of this is with Joseph Meyer so one two three four Maher and myself and five is the same table yes yes I wanted to add some more yeah so yeah I want to add some work I ran out of the board there so I should so of course a few applications well of course step 0 application is where G is a hyperbolic group itself that’s not super exciting and X is the Kayla graph with some generator then of course well this is a nice space ok yes this is finally generated maybe then you get a nice proper space so basically everything I said was basically already known maybe it’s hard to trace back exactly could they do it but certainly group actions on proper hyperbolic spaces were considered certainly my kind my knowledge and then several other people of course you are interested in groups which are not quite hyperbolic almost like the both right so for instance if you have G H which is relatively hyperbolic the group relative hyperbolic the subgroup well you can take X to be the relative space so the count of space and that’s that’s no longer proper space of course so I think they have all expresses still you can take the Bowditch space so it’s so proper so you can do something else perhaps but anyways this is already a case where you have a non proper space then of course they’re the groups were really interested in in topology so one group is the mapping class group and then of course you could take the car complex which we all know is hyperbolic by work of major and Minsky and here very essentially is not proper so you can think of X being some sort of locally infinite graph and you’re going to see what happens for local him photographs but anyways so here again various things were already known by the way it’s also now that the action is a cylindrical here so for instance we know various things so for for instance we know that the the generic element is Sudan also this of course we already knew by work of Meyer and ribbon independently and here we sort of leverage this to more general case where we don’t have only just one element where we consider K elements and we take the group generated business and so so other consequences which you get out of this for the mapping class group is that random K generated subgroup of the mapping class group is complex Co compact because basically well the work of various people including kind of my Nagar is convex compact if and only if it quasi symmetrically embeds in the curve complex so so it does also however note all these groups all these examples

are free are free groups so you get those examples so of course there’s a lot of quest for non free convex compact groups so somehow the random world is too nice you can’t find out any questions another another corollary which of course isn’t it’s known is that well somehow yeah is that you can identify the possible boundary of this width of the mapping class group with the boundary of the curve complex and somehow of course this is already known by work by work of Criminology and Mazur well but then you have to use a lot of machinery to go from here to here somehow I mean in there in original proof you use time or a theory you use the action on Tyco space and then you go back to the boundary of complex and so you also have to prove that it converges in the curve complex then you have a map from the boundary of the curve complex to the boundary segment space to the boundary complet so somehow this proof in a way you can you can go through ways so one you can go to take on a space which is proper but not hyperbolic and in the other case you can go to the third complex which is no proper and hyperbolic and somehow in a way this is more communitarian so if you like if you don’t like taking a theory in a way this is more or less the first proof where you don’t really use any tech north you except maybe to provide cylindrical iti think so far we have to use some machinery but anyways if someone could prove this without any any geometric structures there will be sort of purely combinatorial so I think that’s also an interesting ok so then there’s also sorts of other things but anyways more or less we got the gist also of course other interesting examples are for other thing so if you take G autumn or through some other automorphism groups this free group then ok then you can have a lot of factors so for instance the free factors on books so everything works out nicely so you get yet a lot of interesting you get you get all this so for instance you get a corollary that general generic elements are a troy dough and their fuel irreducible well you need to use some other complex not quite this one but there’s this various variations as you will know but better than me and anyways of course this will was due to copulation and Raven and as I said we can go further from this for instance we can also prove that random free group extensions our hyperbolic and what does this mean well recall that you have the sequence from the full of the morphism group to other fan left well he was almost right that’s good okay whatever so how can you construct a random extension of the free group where you do the following you pick a random K generated subgroup gamma n in outer fan out of maybe should be so we know how to construct around and generate a sub loop here we do K random walks and then you pull it back to here so then you if you have gamma hand here which is here when you have a projection map you pull it back you still have this

so this becomes an extension of the free group and then you have to use a work of Dowell and Taylor to show that if this pie illness of gamma and Qi embeds so if gamma and sorry if gamma and Qi embeds in someone it’s a freshman some sort of three factor complex then this extension is hyperbolic in the sense so some sort of you know involves procedure not too involved I mean you take it take every time you have a group outer automorphism of some group you can take an extension of that group yes yeah also you could do it for other groups but it seems like there aren’t too many interesting groups where the outer automorphism group is is non-trivial business I mean you could do product of surface group times free group we will still work out but then you can’t do much more than that so many questions yes no so the good question is it’s WPD enough and maybe but yeah so we still have we still having I don’t have a full answer to this yet because somehow you want to really look at pairs like a generic pair of boundary elements and see if you can connect them by a geodesic such that there’s not too many elements and so small the stabilizer is sort of has linear growth and so of course WPD is a weaker statement and maybe you can sort of do something but it’s somehow for for the possibility of a defend this wouldn’t be you know unfortunately other questions okay so so now that I I mean I I I took for granted a lot of topology because this is topology same so I don’t know on the other hand I cannot define all all these objects but now we’re gonna go much slower in a way and sort of give an indication of what where is the essential point and in this theory how it’s sort of differentiates itself from the earlier theory when ax is proper and actually fortunately or unfortunately there’s not going to be much topology but it’s more like a gothic theory so so the essential ingredients in the proof so essentially there is one very important ingredient which is the concept of stationary measure so if you have a group acting on some space in its boundary we can say that so if M is a G space new a probability measure on M is new stationary if it’s not quite invariant but it’s invariant on average so you start with new you can push it forward by group element then you’re gonna get quite back new but then you can choose G randomly so if you you can take this is you see you have this probability on you so if you take the average of those according to this probability you get back the same measure okay so that’s the key point and the idea of using special measures goes back to – to the beginning I mean it goes back to work a first time bird and then it’s heavily used in Margulis super rigidity theorem so so why do we use it so here’s the sketch although say you want to proof of convergence to the

boundary if the boundary of X is compact so this would be in a classical case that’s like okay like hyperbolic lane you have the circles it’s compact nice so if I claim that if it’s compact then that if there’s a nice argument which is hidden in first of all get modulus and it’s it’s kind of indirect though it’s using measure theory so here’s the thing so first thing is by compactness so if you have the compactness of the back of this space then the set of probability measures on this face is also compact and then you can apply whatever fixed point theorem you like to say that there exists a stationary measure and the compactness here is absolutely essential now let me say what goes on later because it’s kind of cool so then you use then by the martingale convergence theorem what do you do you take you take your measure which is stationary and now you push it forward by the group elements where your random walk so you consider the sequence W n star you it’s a bunch of sequence of measures on the boundary and by since it’s stationary this sequence is a martingale in the space of measures just sort but anyways this converges to some new so this sequence converges to some measure of course the measure so this depends on the sample pod but anyways this sequence has some limit and now you use the geometry and the geometry tells you the following that by a hyperbola CT if you have a sequence of group elements which converges to to some point on the boundary so this if you have a geometric sequence which converges then there’s also the sequence of measures converges and it converges to a delta mas so to delta methoxy but this is a delta mas tonight and now here and that’s it so I I think this argument is it’s very neat so why wait why does this imply a convergence to the boundary well you consider the sequence G an X now since this is a sequence okay everything is compact so it has to have some limit point it has a limit point by compactness now the only thing you want to exclude is that it has to limit points but the thing is suppose if if it has to limit points what happens well there’s some subsequence G&K x which converges to Delta R see what sorry took c1 and then some other sequence which converges took c2 but now we can use this geometric fact so here we use the geometry to say that this implies the convergence in the space of measures so this would imply the gnk neo converges to this Delta mas this would imply the gmk no converges to this Delta us aha but this is a contradiction because we have the writing a convergence theorem so this sequence converges globally so let’s contradict virgins yeah use compactness is a bunch of places and the real essential one is

the first one so now what is the non proper case in the non proper case as I said the Graham of boundary is not compounded for instance if you think about an infinite tree infinite valence tree something like this like this they called it the infinite Hydra you know there’s so many there’s too many branches so what’s the boundary here well this one boundary point for each ray so here the gram of boundary is vacant and so it’s not compact so if you were to run this argument in the horror function in the realm of boundary then you would run into big trouble because you can’t even start so here’s the workaround so the trick is to consider another boundary you consider what’s called the horror function boundary well maybe other people call it slightly differently but this idea also kind of goes back to grow more or less however there are various definitions which do not coincide for proper case for non proper cases so what is that you consider so if you pick Y in the space you can define a horror function what’s a horror function for us is a function of this form ro Y of Z equals the distance from Y to Z minus the distance from Y to X naught where you pick your fixed base point X naught so that you have the draw Y of X naught is always zero so these functions this you take the distance are all one Lipschitz functions and they all vanish on X naught so you have a map from X this row from the space of C of X also one Lipschitz Lipschitz and f of X continues and also of on Lipschitz is such of f of X naught is zero and now here’s the trick the trick is that this space is also pretty horrible but if you even doubt with the right topology which is the topology of convergence point wise convergence if you equip see of X and well let’s say C of X so decorated that means no no something just our I don’t know so it’s not quite just a continuous function with the topology of point-wise convergence then it is compact it’s just by the ticker of theorem I mean you know you look at the function at each point so it’s like a it’s like a huge product space really huge so that’s not so so the topology is quite weak however this is a good starting point so and not just that well if X is separable this is also mattre instabilities so this means that there exists especially we measure here on on what on the we define the we define a whole function boundary X H to be the closure of the image of this map so however it is this space is pretty weird so let me I have a guest one minute or two so let me just finish by saying two things so first of all this this topology here is pretty weird for instance if you have a sequence of points which go like this yeah so you’re going to infinity I don’t

know where where are you going it’s not clear hello in this weird space there is ooh and there is a limit point and the limit point is the origin and why is that because you see this function is zero here and goes up linearly along the Ray and you know you look at the Ray and you wait until this sequence passes by and once it passes you know you get back the same function which is just linearly increasing along the Ray so somehow you see this topology is pretty weak however the good thing is things converge always now let me just finish we say was the second ingredient just to because now okay we have a stationary measure on on the horror function boundary but what can you do with this of course we want to say something about the growth boundary and so the second ingredient let me just say it and stop is the local minimum up so there is a map from the whole function boundary to X Union the boundary of X and essentially the map is the following the map is the following is this map is G invariant the map is the following you take a photo function and there’s two choices either there is a global minimum so if you so then your map it to X such that X is the minimum or fee okay course leaders but anyways what’s good about this function is this they’re convex and using a hypervelocity you can prove that functions we such that there have a minimum they look like this so they have a 1 minimum or there is no minimum so if the infimum of h is negative infinity then there exists a sequence of course such that H of xn goes to negative infinity by definition then you pick this sequence you pick and you’ll define you pick this sequence to define field H as the limit of this sequence so you have to check that if a whole function doesn’t have a minimum so it looks like must look like this and if you take a sequence of point such that the value of the order function goes to negative infinity this sequence must converge in a growth boundary this is pretty strong bite one huh one can prove that and actually the limit point doesn’t depend so basically so here is the case where the infimum of h is bigger than negative infinity and here’s the case ready fimo the Infinity and so from here you can define a such map and it’s gene variant then of course there’s a lot of work to do but at least this tells us that if you have a session to measure here you can push it forward to a session or a measure here and then you can start running the rest of you there’s still other issues there anyways thank you very much