CONFERENCE
Recording during the thematic meeting : «Discrete Mathematics and Computer Science » the January 30, 2024 at the Centre International de Rencontres Mathématiques (Marseille, France)
Filmmaker : Luca Récanzone
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Okay so this is a second lecture on diagram groups and uh so we will uh use a perspective which will be more geometric so if you remember the definition of diagram groups it was essentially com combinatorial I mean uh in a way which is actually rather
Similar to free group I mean and maybe I will start with some um comparison between diagram groups and and free groups so free groups are are defined by words up to uh free rediction and you the product is just the concatenation of roots con okay and in something in in a similar
Way diagram groups are made of so elements of the group are diagrams you have a notion of dipole rediction and the product of diagrams is just against the concatenation okay so loosely speaking one can say is that diagram groups are like free groups but in two Dimensions um in some sense okay so
Where is the geometry in everything uh so for free groups we know that they act nicely on some trees I mean if you take a free basis then you look at the kaph and you have an action by left multiplication and the cigraph with respect to free basis turn
That to be tree and by using this nice geometry you can uh deduce some useful information about uh the the free group and there is something similar here and so this is not no longer a tree but it’s something which is not too far away it’s a meding
Graph okay and my goal today is to explain how you can con what is first meding graph okay and how you can construct such a graph from a DI diagram group or you can define an action and we can use this action to some useful information on the group okay so maybe I
Will start with some crash GRS on meding graphs okay so maybe to fix the definition for me a graph today will be um is just a set um let’s say a graph x and VX or anyway so it’s just a set with uh a relation which is uh symmetric and anti-reflexive okay
So okay so a set so it will be an element of this set will be a vertex and the relation is uh Define the notion of adjacency okay so in part particular from um a geometric point of view a point of a graph is really a Vertex so I I I do not
Imagine uh edges like being like existing at all they are just relation between uh between vertices and so given graph the distance uh to for all vertices let’s say A and B so the distance between A and B is the smallest length of a path connecting a to
B okay so a pass just a sequence of vertices so that any two consecutive vertices is are adjacent so with respect to this uh relation and um okay so for today um all graphs are connected because I want to do some geometry so I want my metric to be a
Metric and in particular I want this quantity to be finite for any two vertices and if I take of course if my graph is not connected and if I take two vertices in this connected component I have no path between the two and this quantity becomes infinite and it’s not
Something I want today so for today all the graphs will be connected okay so uh now what’s a Medan graph so definition so a connected graph or graph is [Applause] median if for all vertices say X1 X2 and c x uh there exist a unique
Vertex so uh um unique so it is called the median point so what does it mean it means that the distance between x i and XJ is the distance between x i and the median point plus the distance from the Medan point to XJ and this is true for every ient from J
Okay so that’s the definition the the intuition to to have in mind is that we have three vertices okay and what we want is that there exist one vertex and only one such that the distance between say X1 and X2 is equal to the distance between X1 and M plus the distance
Between M and X2 and the same thing for all the three possible configurations [Applause] okay so an example which should be clear is uh trees are meeting graphs because if I take three vertices then typically they will delit a tripod okay and the center of the tripod is exactly the median
Point so this is yes it’s it has to be unique to be median it has to be unique okay so uh the property of being median is also stable under products so if you take a product of median graphs it’s also a median graph so products of trees are other examples of median
Graphs so so this includes for instance the [Applause] grid okay this is an example yes can you show an example where you don’t have sure so not an example so for instance if you take the graph this one so this is k23 so in this case you have uh you take
Uh this uh three vertices and you have two median points this one and this one this is an example of a graph which is not median so in this case of the grid let’s say I take this vertex this one and this one then you can always find uh the
Median so the median will be just you take on each this is the product so on each coordinate you take the median and what you find is so on this Factor the median is just this point and on this Factor the median is just this point so your median point is this
One and you get something like this okay so you also have cubes of arbitrary Dimensions so again they are product of trees I mean they’re just product of edges okay example so if I have just a line graph just one dimension so if if you have
Uh yes okay so on this yes okay on Tree in this case I took three points in in some generic configuration but if they are just on the line then in this case the Medan point is just the sorry the medium one is the one out of the
Three exactly exactly and the grid is the product of two lines so exactly other questions on the definitions no you have examples that are not product of trees sure uh for instance uh you take some you can you can do something stupid you take two examples and you
Glue them so for instance take a cube and you add know an age this is no longer a product and it’s still a median graphes so so okay the cycle so the cycle is not an example because uh so in the list of n examples if you take uh let’s say uh for
Instance I don’t know hexagon then what is the median point of say this one this one and this one there is no m point there is there there is a hole in some sense so for but a for cycle is a median graph right because this is an
Example of of a grid very small grid and this is the the only ex the only cycle which is median so a triangle is not median either not median because if I take the three vertices there is no point in the middle to to be the median Point basically okay yeah question isation
Um not really um okay actually yes um uh okay but it’s a long story so maybe I should not start on this subject but okay but what some something I can say is that a median graph may not be hyperbolic and a hyperbolic graph may not be median okay but there is some
Some from the point of view of last lar scale geometry there is some connection okay okay so speaking about geometry group Theory um a remark because if you come from geometric JP Theory maybe you already met uh graphs but with another name and they’re the same so a cube
Complex is just a space obtained by gluing cubes together cubes of arbitrary Dimensions okay and so the word cat zero means that you can put on such a space a matric which in some sense look looks non positively curved so on each Cube so each Cube has a matric it’s a ukan space
Okay and you extend this metric to the whole Cube complex and you want this metric to be C zero so satisfying some condition I don’t really want to describe today but which means that essentially the space has some flavor of of non-positive curvature okay and so there is a close connection between the
Two and actually the two families of spaces are basically the same so if I take a c Zer Cube complex I can look at its one skeleton and its one skeleton is always a median graph and conversely if I take a median graph then I can I can look at its Cube
Completion which is a cube complex obtained by from the graph by just adding cubes whenever you want you you you can so each time inside the graph you see the one scon of a k Cube then you fill in with with a K Cube okay so you have all the cubes you
Possibly and so the cube complex you obtain is always is always C zero okay so in Geometry group theories the terminology which is the most used is C Cube complex but um uh today I will speak about medium graphs and this is a bit of terminology
Anyway uh okay but but okay so so the the idea to keep in mind is that in some sense median graphs are non positively curved okay this is something which which is useful to keep in mind okay that something really geometric behind Okay um something I would like to say is uh so
This is a definition of median graph some examples uh there is one tool which is really useful to study median graphs it’s a notion of of hyperplanes so I write a definition so uh a hyper plane is an equivalent class of edges uh is a class of it yeah
Okay with respect to uh the reflexive transitive closure of the relation that if you take a for cycle then you identify two opposite edges okay so this is kind of a notion of parallelism okay so I will give an example um okay so let’s draw some more or less generic
Um okay so this is the median graph if I all the edges are here right it’s okay okay so what’s an example of a hyper plane so it’s an equivalent class of Ages so let’s start with some age let’s say this one and what I want to do is to add an
Age in the class each time I see a for cycle and there is an The Edge is opposite to the edge I already have so for instance for this particular Edge I I see a four cycle here so I know that this age has to be in the class as well
But I also have a force cycle here so I also have this age and I keep going so I have a for cycle this age this age this age and that’s it this is an example of H plane right yes uh yes in some sense because you have this equivalence uh
What Cube comp yes but at least Cube complex so it’s a space made of Cubes so in some sense yes you can always obtain so there is also a characterization um so Medan graphs are exactly the graphs that can be obtained from an infinite from a cube possibly infinite uh by
Retraction so there is really close connection between medum grass and cubes and in particular uh you have this this proposition yes so so this Cube complex in particular is Simply Connected so it’s a way to say that if you fill in the cubes you get something Simply Connected so everything is made of
Cubes okay so let’s take another happy plane if I have this Edge I see that I have two Force Cycles here another one another one okay so this is another um I play maybe your last one if I take so for instance let’s say this one an equivalent class with a single age
Okay so you can you see that when you compare distinct Apple planes uh they can in interact in some different ways so oh maybe I need another one yes sorry otherwise I need some let say this this one okay so so the blue and red red hyperplanes are
Transverse okay so what does it mean it means that it’s possible to find a for cycle such that I have two opposite edges in one hyperplane and the other two opposite edges in the other hyper plane okay what I mean by being transverse
Okay uh but I want I also want to to say that if I take the the orange and blue H ples uh so they are tangent so why is our tangent it’s because it’s possible to find two edges uh one in each hyper plane so that
So they intersect they have a common end point but they do not span a for cycle either the F they span the for cycle and this case they are transverse or they do not and in this case they are tangent okay but they have to to intersect so
For instance if I take the green hyper plane it is neither transverse nor tangent to uh all the other hyperlanes so a range red and blue okay so why are uh important it’s because in some sense so um so the geometry of a hyper plane sorry of the meding [Applause] graph is encoded
In the combinatoric of it hyperlanes [Applause] okay so roughly speaking if you understand the have a planes then always they interact okay then you understand how the geometry of your space okay so maybe I will state one state one theorem uh going in this direction to motivate this
Claim so um let X be a median graph then I have the following uh the following statements so first uh for every app Lane so the graph obtain by removing all the ages from my plane obtain by [Applause] so so such a graph has two connected components
Always so they are naturally called Al spaces and they are [Applause] convex so on this example for instance if so by removing the ages I mean that I remove just the edges okay I do not touch the the end points the vertices so if I remove the the orange
Edges I I indeed I I get two connected components one on this side one this one and if I remove the blue edges I again get two connected components okay so that a the first the first point so there is no geometry in it just
With context but so another is so so it makes sense to say that a pass uh crosses or not uh hyper plane so uh a path is a geodesic so it has minimal length geodesic if and only if it crosses each hyper plane at most once and a consequence is that uh for
All vertices the distance between my two vertices coincide with the number of hyper planes separating X and Y okay so I can really recover the metric just from the hyperplanes okay so may let’s consider an explicit example example if I take uh this vertex and this one so X and
Y so the distance between X and Y so in this case it’s easy to compute it’s one two three four five so there must exist exactly five hyper planes separating X and Y so you see that there is a blue one the red one the orange one so that’s three and there
Are two others so you have this one and you also have uh this one here cutting the cube right okay okay so so this is basically what I want to say about median graphs uh now I would like to explain how to construct median graphs from diagram good okay okay so uh two
Okay so um so remember that I told at the beginning of the talk that there is a connection between diagram groups and free groups and how can you how is a tree obtained from free groups it is obtained just by uh so this is a cigraph
So vertex in the in the tree is just a word or an element of the group it’s a r world and the construction will be basically the same for diagram groups a vertex of the graph will be just a diagrams a diagram okay so okay so maybe I just write the
Definition uh first May a statement okay yes a true statement so uh let p r be a semigroup presentation and W A word then so diagram group DPW acts freely on the median graph uh let’s call it mpw okay and now I would like to Define
Mpw so uh before defining mpw I we Define mp uh so let MP be the graph so I want whose vertices are uh the diagrams up to dipole [Applause] redution okay so basically the same thing as for the free groups and now uh with ages connect uh Delta so class Delta to uh
[Applause] Delta conc concatenating with something and where a has a single two cell okay so for three groups the cigraph is obtained by so the vertices are just the element of the group and you move in the graph by right multiplica by some generator right so in
This case so I I I I add just one letter to the to the right in this case I take my diagram okay and what I do is that I just add a two cell at the bottom okay uh so that’s the definition maybe I would give an example right away sorry
Yes what is that two cell so two cell is so in in in the diagrams uh so I will give an example that’s m okay so so let’s consider this semigroup presentation so the same as before okay and now I want to construct this graph
At least a piece of this graph of course the graph is infinite so I just draw a piece of it so I will start with one diagram the simplest POS one simplest possible let’s say that I start with a diagram with no with no cell uh I will
Take the same as in my notes so a b a a okay so now this is just one vertex of My Graph and now the neighbors are obtained by adding two cells so what does it mean it means that I want to apply a substitution coming from my semigroup presentation so I glue
Something below and for instance I will apply uh I need to draw it in the correct way so for instance I have a b a a and I uh I already forgot what I just yeah this one so it’s what I mean by a two cell right is something like this so
It’s one neighbor I I add one two cell at the bottom but of course it’s not the only possible Choice okay I can for instance add this one a b a a uh this one okay or I can add um a b a a uh this one or I can
Add a b a a uh this one and I think that’s it uh no that’s not it okay anyway it’s four possibilities there is one more but you there is otherwise drawing will be really bad uh so you have four neighbors okay so that’s just a little piece of the graph
But now my claim is that of course this will be my median graph right yes so I thought that the vertices are supposed to be diagrams exactly but these are just words right I mean this is a diagram right yeah to they are the same to Define spherical diagrams to define the
Group but a diagram in full generality there is no restriction ver is notes to element of group right and that’s the difference okay maybe I I understand why it’s confusing because for three groups each vertex is an is an element of the group it’s no longer the
Case here it’s a diagram but not not necessarily an element of the group because it’s not spherical you do not have the same top and bottom path right yeah are questions on yeah we keep drawing and you you tell me if there is something not
Clear because so the point is is that it will be actually not really median but almost but there there should be some some some some cubes in it okay and we see that there are some cubes because in this case you glue something here and to go over there you
GLW something here but you can do the two at the same time and what you get is this diagram a a and this one a a and you see that there is a square a four cycle and with this guy if you add you
Will see a cube so a s Cube so a b a a I glue this one and uh this one and there is another one here a b a a and this one you have something here uh something here and the last one when you put everything so you
Have a a a a a a okay and you have a three cube right if you can see it and okay so maybe something I should say uh let’s say I start from this vertex I want to add a new cell right so something I can do is um add a new cell
Uh so so so I want to transform B to AB so what I get is okay okay so that should be a neighbor of this one right but I have dipole dediction and my vertices so I I only draw diagrams but to be precise there are not diagrams
There are classes of diagrams of two diple rejections so this diagram is the same as this one okay so actually when I uh so when I I I add this two cell I’m just going back to my initial Point okay and it’s how I draw my uh my graph
Okay but there is something which is not so it’s not so it’s almost Milan graph but not quite it’s because this graph is not connected typically so there is the fact is that uh so Delta one if you take two classes of diagrams so they are in the same connected
Component of M of P if and only if they have the same top path [Applause] so it should be easy to see but top and bottom are symmetric no no no not in this case because I look at all the possible diagrams not only the spherical
One so in this definition I’m not asking the top and the bottom to be exactly the same in the definition of the group yes but not in the definition of the graph okay so uh so one direction should be clear so uh yes yes yes it’s because my graph is not
Connected so it’s not the graph I want so actually mpw will be a connected component of M of P okay so one direction should be easy it is that if it is in the same connected component then they have the same top path and it’s just because by very
Definition of the graph I just move in one connected component by gluing something below so I do not modify anything above okay so if I have a path from one diagram to another then necessarily the top path should be exactly the same so this is just a the consequence of the definition basically
And for the other way so if you imagine that the two uh so if I so I want to prove that that the the same connected component under this assumption so I need to construct a sequence of two cells such that I can glue them below Delta one to get Delta 2
Yes this isn’t a complete connecting component right sure also change ba into a and the new part of it yeah yes of course so typically the graph will be infinite yeah so I will always draw pieces of the graph okay so uh if so what I want to do is to say that
Delta 1 is Delta 2 times something right I want to to to obtain Delta one from Delta 2 just by gluing something below so by right multiplying by something okay but okay this is just something really stupid to say is that this is Delta 2 minus one Delta 1
And that that that makes sense precisely because I have this condition okay because the concatenation is it makes sense only if the bottom path of the first factor is the same as the top path of second factor and the top the bottom path of Delta 2 is the
Top path of Delta 2 minus one is the top path of Delta 2 so I get exactly this condition okay so that’s the proof of the fact basically okay so now I I I I can finally Define uh mpw so so mpw is the connected [Applause] component of MP
Containing while all the diagrams with W at the top so let’s say one which is the simplest just this one but actually I will obtain all zes okay okay so my theorem tells me that first uh so this should be a med graph and I’m supposed to have an action of
The diag on it okay and and and yes both of action should be clear right so DPW acts on mpw by left [Applause] multiplication okay and this is well defined because so we know that all the vertices of mpw are given by diagrams with W at the top but an element of my
Group is a spherical diagram with W at the top and at the bottom so the left multiplication is well defined that makes sense and the the action preserves the structure of the graph because if I have two adjacent edges it means that um I can glue something at the bottom but
It’s independent of what I do at the top so uh the action sends edges to edges and nonadjacent vertices to nonadjacent vertices okay so um yes so some remark so okay I’m not going to justify why this is true uh why it’s median uh because it requires some work
But it is so DP so mpw is a meting graph uh you have an action which is defined exactly by life multiplication and the action is free so the fact that it is fre is exactly for the same reason in kphs I mean you are just acting by left
Multiplication so if uh you you fix a Vertex then uh you must be the trivial element just by the constellation property so maybe I would like to add two remarks uh because this graph can be uh can be sort of in from different point of view so the first one is that
Mpw uh it coincides with uh the one skeleton of the universal cover of the square complex defined yesterday okay so we saw yesterday that DPW can be is isomorphic to the fundamental group of some Square complex the square complex but now if you take this complex you
Take the universal cover and you only keep the one skeleton you get a graph and this graph is exactly the same as this one okay this is one different point of view and another one which is essentially what I just said but with some big words is that m mpw is a connected
Component of the K graph of so the groupoid of diagrams [Applause] so up to diple reduction always and so with respect to some finite Genting set so to the diagrams with only one cell so that’s just a fancy way yes comp uh okay in this case so spw is a connected component uh
Component of s p uh containing W okay so this is really just a fancy way to say what I just said okay it’s I do not have for fre groups I have a group so I have a cigraph and everything is fine in this case I do not have a
Group but I have something really close I have a groupid which means that I have a product I have inverses I have natural elements but the product is not always well defined that’s the only difference between a group and group otherwise everything made sense so I have a gec
Set which is just a set of diagrams with only one cell and I can look at the kaph with respect to this generating set in the group O the kph is not necessarily connected so I have to choose one connected component I choose a connected component containing the natural element
Let’s say just W so the diagram is no cell and I get exactly the same construction but um I think this is a point of view which is rather which is which is nice it’s it’s interesting okay are there questions on all this constriction okay
Okay so so far we have a group we have a a space we have an action and we’re supposed to be able to extract some information from this data about the group so what can be said about um diagram groups by using these actions so let’s consider some
Applications so it will be essentially um a list unfortunately um so first thing is that um a diagram group with no Z square is locally free so by locally free I mean that every fin generated subgroup is free okay so in particular this prevents many groups from being diagram groups um
So if you take I don’t know your the fundamental group of your surface of genus G at least two so genus at least two then this is not a diagram group because it’s a group is know that square but it’s not free it’s not it is finally generated so it’s not a diagram
Group for instance and it’s possible to prove this statement by using this action on the median graph uh something else so there is some uh some yeah some dichotomy in some sense is that if you have a group a subgroup which is not too far away from
Being ailion then actually it has to be a billion there is a gap in all the possible behaviors of subgroups so every Neil potent and polycyclic I click subgroup in a diagram group is freab [Applause] bilion okay sorry is Al uh not really uh the point is that for instance
You have Thompson’s group F so it is a diagram group it is not uh close from being fre Aban at all I mean and uh it does not contain a free Aban subgroup so the precise T alternative is not satisfied by diagram groups but now there may be other other types of
Alternative which may hold but I don’t know Any uh so I don’t really know okay so um there some example for instance so let’s say so the Eisenberg group for instance so not diagr group okay uh so you have this statement for nil Po and polycyclic groups but you can you can do have something better I
Mean you have metab billian groups that are um uh diagram groups so for instance you take the risk product with Z with Z so this is a diagram group and is met [Applause] metabion so the point is that if you are with close foring ailan you are million but
There a really Gap but uh you have also examples which are considered as not being too far away from a million and which are not a million but are example of diagram groups okay uh third point so uh if p is a finite [Applause] presentation uh of a finite sem group [Applause] then
DPW uh is of type f Infinity so what does it mean it means that uh it admits uh classifying space uh with Finly many cells in each [Applause] Dimension okay so this property is usually called the finiteness properties because uh so your group uh your group is Finly generated
If and only if it admits a classifying space with finally meaning one cells it is finally presented if and only if it admits a classifying space with finally many one cells and two cells and so you can keep going and eventually you get something really strong fin really strong and you can
Deduce from the action on DP of DPW on so not exactly actually the the median graph but that in on on its Cube completion and by using some argument coming from more Theory you can deduce that there exist such a classifying space okay and for instance this property this statement
Applies to Thomson’s group F okay remember that in this case P the presentation is just uh one generator and the relation is X = x² right so this is a finite presentation of finite sem group so from this statement it follows that f um is of type f
Infinity all right but there are not all diagrams all diagram groups are of type f Infinity for instance if you take uh so this one this one is finally generated but not finally presented so it it it does not admit such a CL classifying space actually it does not
Admit a classifying space with fin many one cell and two cells so it’s specific to one particular family of diagram groups okay another one so if the class of w is finite so by the class of w I mean I mean the set of all words equal to W in the
Semigroup okay so if this class is finite then DPW is a subgroup of a right angle arting group and for instance it is linear over Zed okay uh do I need to write the definition of a rag yes okay okay so definition of a
Rag H um let’s say gamma is graph so not necessarily connected so actually when I say that all graphs are connected they I so let gamma be a graph then the right angle arting group a gamma is defined by the following presentation so the generators are just the vertices and the relations are
Commutators and I want to say that two adjacent vertices commute okay so the idea to keep in mind is that if uh there is no age aamma is free right because I do not have any relation so my group is just a f group if gamma is complete then aamma is free ailon
Nice so if gamma is complete it means that any two generators commute and so I get an Aban group and usually what we say is so you have two extreme situations when you have no commutation you have free group when you have commutation everywhere you have three
Aban group and everything is between is just a fam of Rags so it’s some sort of of an interpolation between free groups and free Aban groups okay and okay and so are very nice groups and uh so what the the claim here is about is if I take if I have this
Assumption that the class of w is finite then my group will be isomorphic to a subgroup of right angular group okay so in particular all the nice properties I know about Rags and that pass through subgroups will be automatically Satisfied by these diagram groups and in particular being linear
Over Z okay so being isomorphic to subgroup of a Matrix Group uh with Z coefficients and again it’s true only for some diagram groups so for instance again Thompson’s group f is not a sub group of a rag because it is for instance it is not linear it’s not even resid residually finite
Um okay maybe another one and a part of this statement will be proved in the exercises I mean if you have a group acting on median graph you automatically have a morphism from the group to a right angle so not AR groups but Co groups but just add a relation
You put U squal one you always have a morphism and uh so of course in general this morphism is Trivial you don’t you cannot say anything in full generality but in this under this assumption you can use a specific property but the action to deduce that this morphism will be injective okay
Five so every diagram group satisfies a short exact sequence uh let’s say G okay so you have a short exact sequence and what is Q it is a subgroup of Thompson’s group F and what is n it’s a subgroup of a right angular group so this statement tells you that in some
Sense every diagram group can be constructed from subgroups of rags and subgroups of Thon group F okay so this is a very uh strong decomposition on the other side uh the class of subgroups of Rags is really huge we do not understand them them and the class of subgroups of Thon group f
Is really huge and we do not understand these groups so it’s an interesting statement but it does not mean that we understand everything about diagram groups just from this chart exact sequence uh something interesting is that it allows you to construct subgroups of f because if you have some
Diagram group and you are able to to show that this Kel must be trivial then you have an embedding inside F and you can construct some non-trivial subgroups of f by using this this short exact sequence so this is a really really nice statement anyway uh okay maybe another one so
If p is a finite presentation then uh uh DPW satisfies the agu property so it means that there exists a proper action by isometry on a Hillar space okay so it’s a property that basically uh tells you that there is some compatibility between the geometry of the group and the geometry of Li
Spaces okay that you you have proper action on H on on HB space and you you also have to there also some other motivation I mean I properties also imply some conjectures with big names that virtually nobody understand but um okay so you have this name but formally it’s really this I
Mean in the intuition is really that you have some compatibility between the geometry of the group and the geometry of leader spaces okay so do I have time for something else okay uh okay so so we talked about diagram groups and but there are also some generalizations so um okay theorem
So Rachel mentioned that there also other types of diagram groups called symmetric and braided uh so symmetric diagram we say respectively uh [Applause] braided so the ACT on the median graph on median graphs uh with finite respectively red groups stabilizers okay so for so maybe I
Should put an example of a diagram so let’s I consider always the same presentation it’s convenient so for for for example an example of of I have a b a a so using the formalism of of transistors what what I can what I can
Do is something like this so I have ba a so the transistor corresponds to applying a relation for my semi representation okay so a becomes ba a but uh and I can for instance also add this one okay so a square become a and I can just
Permute um so this is a symmetric diagram and you can do something similar by by braiding the strength so for instance something like this okay and this is a braid diagram okay so uh so for this generalization s you can also so modify a little bit the construction to get an
Action on median graph but it’s no longer free for symmetry groups the stabilizers so they no longer trivial but they are finite which is not too bad and for Brady diagrams you get an action so in this case the stabilizers become infinite but they are well known they are isomorphic two bread
Groups so they’re infinite but you understand them which is not bad either and uh so maybe one consequence of this stabil uni no exactly they’re not un uniformly finite and this helps me for the transition actually so a consequence of this statement is that uh in uh symmetric diagram groups uh
Torsion fin generated torsion group torsion subgroups are finite so which is not obvious precisely because usually you do not have a uniform bound on the size of finite subgroups so the typical example is when you take a presentation for Thomson groups so in this case you do not have
Uh F anymore you have Thomson’s group V but you can see from this example that you have a lot of of big symmetric groups because if you take uh you you so so you you you apply presentation you apply your relation xal X Square again again and again and again
To have many rers now you do some some permutation and you you just do the same thing again but in the reverse way and when you take uh so this is a subgroup isomorphic to uh to the Symmetry group so in this case it say4 but in general can get a copy of
SN for arit large n okay so in particular for uh so V which is DPX you obtain arbitrary large subgroups and actually you can find a copy of an infinite torsion group uh s Infinity basically uh but if you restrict your attention to finitely generated dorsan subgroups then they must be
Finite and they must be finite because you have uh this action and this median graph is specific so the idea is that in this median graph you do not have infinite cubes you may have you may have cubes of arbitrary large dimensions but nowhere you have an infinite Cube so if you have
A sequence of cubes of increasing Dimensions the dimension the cube has to go to Infinity okay but if you fix one vertex and you look at and you look at what happens around you never see an infinite Cube and with this assumption this is the theorem not at all obvious but if you
Take a finitely generated torsion subgroup you make it act on such median graph and you must stabilize a cube and because the stabilizers are finite you conclude that your group must be finite and so you cannot find an infinite suchar subgroups okay and so maybe one word about uh the construction so the
Construction is basically the same but you have to be careful about permutations of wires okay so the only modification is so vertices are uh diagrams diagrams mular diaper ruction and you are you also allowed to uh permute the bottom wires okay and permutation of bottom wires so for instance
If so again I get the same I take the same presentation if I consider um okay so this is a diagram and if I take the class of this diagram modulo typle reduction and permutation of bottom wires so this will be this will represent the same vertex as this
One and this is the only modification which is needed to uh make the construction work so to get a graph which should be median you take exactly the same definition more or less the same proofs um it’s slightly more complicated but it works and you get a meding graph
With an action and you have a non-trivial Vertex tab iers because uh because precisely of this permutation they are finite because you are allowed to permute the the bottom wires and so the stabilizer of this vertex will precisely a copy of of the symmetric of uh the symmetric group of of three on
Three elements which corespond exactly to the three bottom wires okay so it explains and for bre groups same intuition uh so permutation so you just braid the bottom wires do the same thing again it works you get an action on M graph and the stabilizer of vertices corresponds to uh
To the breid groups on the bottom wires okay so basically the same idea um okay so maybe I will stop now I have nothing else to add I guess uh oh I can add something but maybe not not reasonable no no no I I will stop now thank you
It works so do you have questions for Anthony we already asked some but so I Rush with the microphone um so you said in the beginning that the action of your diagram grip and your median graph is free mhm when is it co compact so it’s
Co compact PR easy to WR no so it’s co compact precisely when uh this class the class of the world is finite So Co compact if and only if uh the class the class of the word is finite so this is a set of all the
Possible words equal to W in the semi group so for instance for this one it does not work because you have a which is equal to a square a cube A4 etc etc and so is not uh the action is not compact but for instance if you remove this relation
Then it becomes compact because you can only just P mute lators and it works and and yeah was that question yeah it’s in so you said that the the cigraph the the median graph is the cigraph of the group oid can we see the cigraph of the group in the median graph
Uh I don’t think so because the problem is that of course the generators are not generators of the group uh and I not sure you can see it from um so not obviously I would say no but I’m not sure I mean it’s difficult to say that something does not exist of
Course uh but my my intuition is that it’s not the case other questions maybe I have two questions just a naive one can there be torsion in the diagram group uh no no uh because of of the fact that no it’s not written so it’s due to the fact that the action is
Free basically uh so free not on okay to be precise the action is more than free you have that Cube stabilizers are okay so uh so the action is not only free on the vertices also free on on the cubes and now if you take a finite for the
Same reason as before if you take a finite subgroup it will stabilize a cube and so it will be trivial because stabiliz the cube is trival and another question your your your short exact sequence mhm uh does it have something to do with ground Ro theorem of decomposition of semi
Groups not I would say no at least it’s not proven during using this uh uh thism but maybe so um how can you say a few words about the orderability either left or by now that you have discussed these median median graphs maybe do you see the so
What is the action on the I’m interested in the perspective of action on the real line mhm so do you somehow get a line somewhere in this graph or okay so I can say many two things uh the first one is so we know that groups are biod durable
But not in an explicit way do not have an explicit order on all these groups what we know is that so there exist some diagram group uh let’s say U for Universal um such that um so for every diagram Group D quable so D is a subgroup of U but now U
It’s possible to decompose it as a semidirect product between Rag and F it’s actually quite related to this short exact sequence it’s the proof of this short exact sequence is pro using this kind of statement and now this group is bi orderable this one is and you have to check that uh it’s
Possible to extend this these by orders uh through the semidirect product which is not obvious but the decomposition is completely explicit and you can do it it’s possible so it’s not entirly satisfying uh there is another um point of view when you can construct explicit actions on on on the line is so
You have your your your diagram so uh let’s say it’s a a and you have a spherical one let’s say ba a and uh a a squ and a again a a a b a a okay and now what what you can do is you can imagine
Um this line as a copy of 01 this line has another copy of0 one and now for each two cell you will fix homeomorphism from so the sub interval to this interval and you will push it to get theomorphism to Z1 to Z1 and so for instance for toson group F
There is a natural uh so this is X x and x so natural map is so you have morphism from 01 to 02 so something that you to do is just a multiplication by two and when you can when you perform this construction to ter group F you recover its natural action on
01 so you have uh such constructions however there exist diagram groups when it’s not possible to get a faithful action on the line from this construction so it’s more explicit than this approach but it does not work all the time okay so may maybe we stop here
Because the next talk by Nicholas barar is in two minutes so we have a very short break and then we will have a real break after Nicholas talk thank you very much [Applause] Anthony