This thesis takes Brady’s construction of s for the braid groups as a starting point. It is widely known that this construction can – with the right ingredients – be generalized to Artin groups of finite type. Results of Bessis as well as Brady and Watt are used to establish the general construction for Artin groups of finite type. The non-crossing partition lattice in finite Coxeter groups is identified and used to generate the so called poset group. With the help of this poset group, which turns out to be isomorphic to the Artin group, a simplicial complex is constructed on which the poset group acts. It is shown that the complex itself is the universal cover of the of the Artin group of finite type and the quotient under the action is the desired .
Deutscher Titel: Über Bradys klassifizierende Räume für Artin-Gruppen endlichen Typs
Diese Thesis nimmt Bradys Konstruktion von -Räumen für die Zopfgruppen als Ausgangspunkt. Es ist allgemein bekannt, dass diese Konstruktion mit den richtigen Hilfsmitteln auf die Klasse der Artin-Gruppen endlichen Typs verallgemeinert werden kann. Es werden Ergebnisse von Bessis sowie Brady und Watt benutzt um die Verallgemeinerung auf Artin-Gruppen endlichen Typs möglich zu machen. Nichtkreuzende Partitionen in endlichen Coxetergruppen werden identifiziert und benutzt um die sogenannte Poset-Gruppe zu erzeugen. Diese ist isomorph zur Artin-Gruppe und ist Basis für die Konstruktion eines Simplizialkomplexes auf dem die Gruppe wirkt. Es wird gezeigt, dass dieser Simplizialkomplex die universelle Überlagerung des der Artin-Gruppe endlichen Typs ist und dass der Quotientenraum der Gruppenwirkung der gewünschte ist.
On Brady’s Classifying Spaces
for Artin Groups of Finite Type
March 6, 2018
Advisor: JProf. Dr. Petra Schwer
[1cm] Department of Mathematics
[1cm] Karlsruhe Institute of Technology
Acknowledgements.I am very grateful to Julia Heller for many hours of interesting discussions and valuable advice I got on my way to understanding the topic and writing the thesis. She has always been a patient listener to my concerns and provided me with guidance in a kind and dependable way. I greatly appreciate her help and support I received during the process.
- 1 Introduction
- 2 Definitions and Notions
- 3 A Lattice in the Coxeter Groups
- 4 Poset Groups
- 5 Construction of the Classifying Space
- A Topological facts
List of Figures
- 1 Hasse diagram of
- (a) The subcomplex is highlighted.
- (a) The subcomplex is highlighted.
- (b) The subcomplex is highlighted.
- 3 Coxeter graph of
- 4 The generator of .
- 5 Hasse diagram of the reflection order on
- (a) A non-crossing partition
- (a) A non-crossing partition
- (b) A crossing partition
- 7 Hasse diagram of the relevant subset
- 8 Hasse diagram of the subposet from Lemma 4.3
- (a) First part of the induction base
- (a) First part of the induction base
- (b) Second part of the induction base
- (c) Induction step for
- 10 Order complex of the symmetric group on elements
- 11 Gluing to each element of the order complex
- 12 The copy of at and are glued together
- 13 Building the from the order complex of
The class of Artin groups arises from a generalization of the braid groups, which were introduced by Artin in . In this work, Artin proved a certain finite presentation for the braid groups and solved the word problem. Other fundamental work on braid groups includes Garside’s , where a new solution to the word problem is given and the conjugacy problem is solved. Brieskorn and Saito  and Deligne  independently generalized the braid groups to what is known today as Artin groups.
The aim of this thesis is to describe Brady’s construction of s for the braid groups from  and elaborate on the general case of Artin groups of finite type.
Let be a group and a connected topological space. Then is called an Eilenberg-MacLane space of type , if
Instead of Eilenberg-MacLane space of type we will also call such spaces just s. In case is a discrete group, spaces are also called classifying spaces.
Some of the results of  are closely related to Birman, Ko, Lee . Krammer described independently from Brady in  the construction of the very same for the braid groups. That is why in the literature this complex is sometimes called Brady-Krammer complex.
In  Brady and Watt generalized this construction to Artin groups of type and . In the same work they noted that the construction can be generalized to every Artin group of finite type for which one can show that the closed interval forms a lattice in equipped with the reflection order, where is the identity in the related Coxeter group and a Coxeter element. Bessis independently obtained similar results and already established this lattice property for all finite Coxeter groups with a case-by-case proof, that was partly achieved by computer, in . Brady and Watt then gave a case-free proof of the lattice property for all finite Coxeter groups in .
The main goal of  was the study of the dual braid monoid of Artin groups of finite type. This approach can be understood as a dual theory of the positive braid monoid, which was for example studied in . Instead of the positive braid monoid, which comes from the Coxeter group with its standard generating set , the monoid which is generated by the set of all reflections is considered. Bessis introduced this notion of dual Coxeter theory, of which the construction in this thesis also makes use, in . The case when is of type was considered earlier in .
The thesis is structured as follows.
In Section 2 basic definitions are made. Then, in Section 3, the reflection order on a Coxeter group is defined and we observe the non-crossing partition lattice described by Brady and Watt in . We also prove a few lemmas on the structure of the lattice. We then define the poset group for a finite Coxeter group in Section 4 and use a result of Bessis  to establish an isomorphism to the related Artin group. We establish cancellation properties in the positive semi group and show that it embeds into the poset group. In Section 5 we construct a simplicial complex and show that it is the universal cover of the desired .
We describe the combinatorial construction of the Brady-Krammer complex, closely following  and give the general approach for Artin groups of finite type. We use the main result of  to establish the lattice property and make use of a result of Bessis , where he showed that the poset group of a finite Coxeter group – defined in Section 4 – is isomorphic to its related Artin group. We try to elaborate, give explaining examples and go into detail in the proofs so that the construction becomes understandable and easy to read.
2 Definitions and Notions
For the most part, we will follow the notation of Björner and Brenti .
2.1 Partially Ordered Sets
A partially ordered set – or short poset – is a pair of a set and a relation on with the following properties. For all it holds:
if and , then (transitivity),
if and , then (anti-symmetry).
If the order relation is clear from the context, we call a poset. If but , we also write . A cover relation is a pair such that there is no with .
For we call a subposet of when inherits the order of . For we define the interval . A sequence of elements of is called a chain if , where is called the length of the chain. The supremum of the lengths of all chains of is called the rank of . A chain is maximal if its elements are not a proper subset of the elements of any other chain. We call pure if all maximal chains are of the same finite length. We call an element maximal if there is no with .
For a pure poset we define a rank function by letting, for , be the rank of the subposet .
Let be a poset and . We call the meet of and and write if
for all with and it holds .
Analogously, we call the join of and and write if
for all with and it holds .
The meet is the greatest lower bound, while the join is the least upper bound. Meet and join are – if they exist – necessarily unique. If in a poset for all pairs of elements there exists a meet and a join, we call it a lattice.
The Hasse diagram of a poset is a diagram that represents its structure. It is a graph on the vertex set and for each cover relation there is an edge going upwards from to .
Let with partial order relation
The Hasse diagram of is depicted in Figure 1.
2.2 Simplicial Complexes
A non-empty family of finite subsets of a set is called abstract simplicial complex, if for any and it also holds .
The set is called the set of vertices. The elements of , which are subsets of , are called faces. Note that since we require an abstract simplicial complex to be non-empty, we always have . Furthermore, we only consider abstract simplicial complexes where for each vertex , is a face. In this case we identify with the vertex .
The dimension of a face is defined as . The dimension of is defined as the supremum of the dimensions of all faces of . For , we define the interval .
A subset of which is an abstract simplicial complex itself is called a subcomplex of . The specific subcomplex for a face is called a simplex.
For , we define the -skeleton of to be the subcomplex consisting of all faces of dimension at most . Faces of dimension are also called edges.
If is a poset, then we can associate to a specific complex .
For a poset , let be the following set.
The elements of are just all finite chains of . Since all subsets of finite chains are finite chains themselves, is closed under containment and therefore is an abstract simplicial complex.
The complex is called the order complex of .
Let be an abstract simplicial complex, a vertex of , a face and a collection of faces of .
The closure of , denoted as , is the smallest subcomplex of that contains all faces of , i.e.
For a face , the closure is defined as .
The star of , denoted as , is the smallest simplicial complex that contains all faces of which contain as a vertex, i.e.
The link of , denoted as , is the set that contains all faces of which do not contain , i.e.
The (simplicial) cone over , denoted as , is obtained by introducing a new vertex to and adding for each face of , the face .
To specify the cone vertex of a cone over the complex , we sometimes write .
Note that unlike other authors, we define the star as a closed (under containment) subset. Thus, it is always an abstract simplicial complex itself. The link of a vertex is also always an abstract simplicial complex.
Let . Then is an abstract simplicial complex. In fact, it is the order complex of the poset from Example 2.2.
The closure of is .
The star of is .
The link of is .
The cone over with cone vertex is
The following lemma will be needed in the proof of Theorem 5.5, but since it holds for any abstract simplicial complex, we will prove it now.
For an abstract simplicial complex and a vertex of , it holds
Let be a face of . Then there are two cases.
Either contains as a vertex. Then, is a face of , but does not contain . Hence, we have . But then, by definition of the cone, must be a face of .
In the second case does not contain . Thus, we have by definition of the link . But then is also a face of the cone over .
Now let be a face of . Again, there are two cases.
Either is a vertex of , in which case is a face of .
Or is not a vertex of . Then, is already a face of . But then, since the link is a subset of the star, we have .
For any abstract simplicial complex there is a topological space related to it. Such spaces are called geometric simplicial complexes and are the geometric counterpart to abstract simplicial complexes.
A geometric simplex of dimension is the convex hull of affinely independent points in for . The affinely independent points are called the vertices of the geometric simplex and the convex hull of any subset of the vertices is called a face of the simplex.
A geometric simplicial complex is a non-empty collection of geometric simplices, such that
any face of a simplex of is a simplex of and
two simplices of intersect in a common face.
Note that since we defined a face of a simplex to be the convex hull of any subset of its vertices, the second condition could also mean that the intersection of two simplices is empty.
From any geometric simplicial complex we can derive an abstract simplicial complex by taking the sets of vertices of the geometric simplices of to be the faces of the abstract complex. In fact, any abstract simplicial complex can be obtained this way.
The other way around is more interesting for us; from any abstract simplicial complex we can derive a geometric simplicial complex, denoted as , called its geometric realization. This geometric realization is not unique, but all geometric realizations – regarded as topological spaces – are the same up to homeomorphism. Therefore we will talk about the geometric realization. It is fully determined by the combinatorial properties of the abstract simplicial complex.
A sketch of one way to construct the geometric realization of a given abstract simplicial complex is the following.
Take for every face one geometric simplex of dimension , and define a map which identifies the vertices of the face of the abstract complex with the vertices of the geometric simplex . Then, define for each pair with the inclusion of the corresponding geometric simplices, such that the following diagram commutes.
F_1 \arrow[r, "f_F_1"] \arrow[d, " "’] & s_F_1 \arrow[d, hook, "ι_F_1F_2"]
F_2 \arrow[r, "f_F_2"] & s_F_2
Let be the coarsest equivalence relation on such that for all it holds for with .
is the geometric realization of .
Other approaches to defining the geometric realization of an abstract simplicial complex can for example be found in Chapter 1, §4 of .
Some geometric realizations of subsets of the abstract simplicial complex from Example 2.6 are the following.
Throughout this thesis we will often talk about ‘the simplicial complex’ without specifying on whether we mean the abstract simplicial complex or its geometric realization. It should be clear that whenever we refer to combinatorial properties, we mean the abstract complex and when we refer to topological properties, such as contractibility, we mean its geometric realization.
2.3 Coxeter Groups and Artin Groups
A matrix , , with is called a Coxeter matrix if it is symmetric and if and only if .
A group is called a Coxeter group if it admits a presentation with generating set and defining relations , consisting of
of an Coxeter matrix .
is called a Coxeter system with Coxeter group and Coxeter generators . The cardinality of is called the rank of , denoted as . For a Coxeter group , the generating set is not unique. Thus, we have to specify one set of Coxeter generators. The s on the main diagonal of a Coxeter matrix just mean that the generators are all involutions. If an entry of the Coxeter matrix is , this means that the product has ‘order infinity’ – meaning for all , where denotes the identity in .
Another way of describing the relations of a Coxeter group is by the Coxeter graph. The vertex set of the Coxeter graph of is and there is an edge joining and if . If , the edge joining and is labeled by .
Note that we could also write the relations of a Coxeter group as which includes the . But because of the following definition we prefer the former presentation.
Let be a Coxeter system with Coxeter matrix . The Artin group associated to is defined as
An Artin group is said to be of finite type if its related Coxeter group is finite.
Consider the Coxeter matrix . The equivalent Coxeter graph is depicted in Figure 3.
The Coxeter group determined by the Coxeter matrix is the symmetric group , which consists of all permutations of the set . To denote permutations, we use the cycle notation with commas separating the elements in the cycle. A set of Coxeter generators is given by with , the set of all adjacent transpositions. Then the relations are , if and if .
The Artin group associated to is the braid group on strands, . It consists of all braids on strands and is generated by the braids that swap two adjacent strands, left over right. The generator is depicted in Figure 4, as an example.
Since the Artin group which is associated to the symmetric group is the braid group on strands, Artin groups are also called generalized braid groups.
Essentially, the Artin group has the same relations as the Coxeter group except that the generators are not self-inverse. Indeed, we even have for and all , where denotes the identity in the Artin group. To see this, we can construct a homomorphism to . For this, consider the map that maps all elements of to . Then, one can show that this extends to a homomorphism . Since the image of is never for , we know that if .
In particular, no Artin group is finite.
There exists a natural surjective homomorphism which maps each generator in to its counterpart in ,
The kernel of this homomorphism is generated by the set .
In case of the symmetric group and the braid group this homomorphism has a nice depiction. By forgetting how the strands of a braid cross and only looking at the positions of the starting points and end points of the strands, it can be viewed as a permutation. The kernel is then all braids where for each strand starting point and end point are at the same position. Such braids are called pure braids.
The following definition introduces elements of which play a special role. Note that we only define them for finite Coxeter groups .
For a Coxeter system of rank with finite Coxeter group , an element conjugate to is called a Coxeter element.
Note that our definition follows Armstrong , while many authors, as e.g. Humphreys in , define a Coxeter element to be an element of the form , for a permutation in the symmetric group . Since Humphreys showed in Proposition 3.16 of  that any two Coxeter elements are conjugate (for his definition) it follows that our definition includes those elements. From this fact also follows that our Coxeter elements form a single conjugacy class.
3 A Lattice in the Coxeter Groups
3.1 The Reflection Order
From now on we only consider finite Coxeter groups .
For a Coxeter system we define to be the conjugacy closure of . We call the set of reflections and an element a reflection. An element of is also called a simple reflection. For we call with a -decomposition of .
This notion was first introduced by Bessis in . He called a dual Coxeter system and was one of the first to study Coxeter groups with a larger generating set which is closed under conjugation.
For let be the minimal number of reflections in a -decomposition of , the reflection length of .
A -decomposition of an element using reflections is called a reduced -decomposition or – if it is clear from the context – reduced decomposition.
The reflection length of is exactly the geodesic distance of the identity and on the Cayley graph of with generating set .
The reflection length is a conjugacy invariant, i.e.
Let and be a reduced -decomposition of . Then and
Now, since for all and is closed under conjugation, we have for all . Thus, admits a -decomposition with reflections. This shows .
To see , note that is a conjugate of , so the roles can be swapped. ∎
The reflection length is sub-additive, i.e.
Let and , be reduced -decompositions. Then and is a -decomposition of and we have . ∎
Equality holds whenever lies on a geodesic from the identity to in the Cayley graph of with generating set , which in turn is the case if and only if there is a shortest -decomposition of with a -decomposition of being a prefix.
Now we define the partial order that will play a central role in the construction of the s for Artin groups of finite type. Since it is based on the reflection length, we call it reflection order.
Let be the reflection order on defined as
In the literature, the reflection order is also called absolute order.
It is now easy to observe the following lemma.
The group together with the reflection order is a partially ordered set.
We will show that the three defining axioms reflexivity, transitivity and anti-symmetry hold. For this, let .
Since , we have .
If , we can deduce
The inequality is due to the sub-additivity of the reflection length (see Lemma 3.3). But for the same reason we have , which gives equality and therefore .
If , we have . But since the reflection length is non-negative we can conclude . Thus, we have .
3.2 Non-Crossing Partition Lattices
From now on let be any finite Coxeter group, the reflection order on it and an arbitrary Coxeter element in .
Now, as the heading of the section suggests, we want to find a lattice in the poset . Surely, itself is not necessarily a lattice, since two elements of maximal length have no join. That the reflection length on is indeed bounded above was first shown by Carter in [14, Lemma 1-3]. He proved that a maximal element has a reflection length of . Although he only showed this for Weyl groups, the same arguments hold for finite Coxeter groups in general, as noted by Dyer , Bessis , Armstrong  and many others.
It also follows from Carter’s Lemma 3 in  that Coxeter elements attain this maximal length.
Consider for example the symmetric group of all permutations of the three-element set with Coxeter generating set , the set of adjacent transpositions of . The generating set is in this case and the Hasse diagram of the reflection order is displayed in Figure 5.
Obviously the elements and are never comparable, since they have the same reflection length of . In particular, they do not have a join.
However, this example might be misleading. Armstrong noted in  that although all Coxeter elements are maximal elements of the reflection order, in general not all maximal elements are Coxeter elements. The second implication only holds in the case of type Coxeter groups, which are the symmetric groups.
Theorem 3.7 ([11, Theorem 7.8]).
If is a finite Coxeter group equipped with the reflection order and is a Coxeter element, then is a lattice.
The lattice is also referred to as a non-crossing partition lattice. It is an algebraic generalization of the classical non-crossing partitions, which were first studied by Kreweras in 1972 in . He also proved that they form, ordered by refinement, a lattice. Biane proved in  that in the case of type Coxeter groups, the lattice coincides with the classical non-crossing partitions, which can be imagined as follows.
Take the set and place the elements on a circle, circularly ordered in the natural way. Then the non-crossing partitions of this set are precisely those partitions for which one can draw all partition blocks as convex sets such that no two blocks intersect. For some examples of crossing and non-crossing partitions of the set consider Figure 6.
For this reason we will write for the lattice of generalized non-crossing partitions , for a finite Coxeter group and a Coxeter element .
For two Coxeter elements in it holds that
is a poset-isomorphism.
For two Coxeter elements , there is such that , since any two Coxeter elements are conjugate. Then, for it holds
Thus, conjugation with maps the elements of bijectively onto . That this is indeed order-preserving, can be shown in the exact same manner. Since the same holds for the inverse – conjugation with – the lemma is proven.
Because of the previous lemma, we know that the isomorphism type of the non-crossing partition lattice is independent of the choice of a Coxeter element. Therefore we will only refer to it as .
The following results relate the group structure of to the poset structure on given by the reflection order and are taken from , in which Brady constructs the s for the braid groups.
Let with . Then and .
Suppose and let and . Then we have . Since reflection length is a conjugacy invariant, this implies . Therefore, .
For the second inequality note that and . This gives , which is by definition .
Let with . Then and .
Let and . From and we get
From this we get . Also, we have which gives with . We conclude that , which is by definition . By Lemma 3.9 we then have . Thus, by definition of and , it holds and . Thus we have established and , as desired.
The following lemma will be used in Lemma 4.4 and 4.5 to help establish cancellation properties in a semigroup we are about to define in Section 4. It shows that the intuition of the structure of a lattice can in fact be transferred to the group structure on the lattice .
Let and define the elements by the following equations
Then it holds
The situation is depicted in Figure 7.
That the elements are indeed uniquely determined by the equations above follows from the group structure. To see this, note that if , then we have and hence . Therefore, there is exactly one element with .
We only prove the first equation, the other two are similar. We know that and . Thus, , which is equivalent to .
What is left to show is that this is exactly the join of and . For this, note that by definition of the join we have . Then we can apply Lemma 3.10 to deduce that , which is . In the same way we get . From this follows that .
Now assume, for the sake of contradiction, that , say and define the elements by the equation . This is again depicted in Figure 7. Then it holds , from which with Lemma 3.10 follows that . This is equivalent to . We want to show that then is a smaller upper bound for than , which contradicts the minimality of the join of .
We know that and therefore . From the sub-additivity of the reflection length it follows that
Since is true by the sub-additivity of , we have established . Therefore is an upper bound for and with that, one for and . Applying this argument onto instead of gives . This contradicts the minimality of and thus we have shown .
4 Poset Groups
From now on let be any finite Coxeter group and a fixed but arbitrary Coxeter element. Knowing that forms a lattice, we will construct the so called poset group. We will follow the construction of Brady from  which is also used in  from Brady and Watt.
Let the poset group be the following group.
For each element we take one formal generator . The group relations are of the form whenever in and .
In other words, for every relation in the lattice we have a relation in .
We do not explicitly define what is, but it will sometimes occur when we consider words in the generators which represent elements of . In this case we regard it as the empty word.
The group constructed here is exactly the group which Bessis called in .
Starting with this poset group we construct a . To obtain a for the Artin group, we will then show that the related Artin group is isomorphic to the poset group . Therefore, the following theorem, which is taken from , is crucial for our construction. Bessis uses a case-by-case proof, which is partially achieved by computer. Up to now, no case free proof is known for this fact.
Theorem 4.2 ([4, Theorem 2.2.5]).
Let be a finite Coxeter group and a Coxeter element. Then
4.1 A Cancellative Semi Group in the Poset Group
Since the relations that are used in the definition of do not involve inverses of generators, we can use the very same presentation to define a semigroup . The following definitions and notions were introduced by Garside in , who used them to give a solution to the conjugacy problem in the braid groups.
A positive word is a word in the generators of that does not involve an inverse. Positive words represent elements of . Two positive words are positively equal if there exists a sequence of positive words , such that and is obtained from by replacing one side of a defining relation by the other. If and are positively equal we also write .
If two words and in the generators of are identical letter by letter, we write . If two positive words are identical, they are also positively equal.
The reflection length on elements of can be used to associate a length to each generator of . And since the semigroup is only defined by relations of the form whenever in and , the relations in also preserve the length and we can associate to each positive word a length . For the same reason two positively equal words must have the same length.
Our next goal is to establish cancellation properties in and then show that embeds in . Therefor we first need some lemmas.
Let with and for . Then .
Using the same arguments on we can show that and . But then we have found an element which is greater than and but strictly less than the join of and . This contradicts the minimality of the join of and .
Let and be positive words in .
If then there exist and a positive word , such that
Following the proof of Brady in , who himself refers to Garside , we do double induction on firstly, the length of – which is the same as the length of – and secondly on the number of substitutions in the sequence of positive words which realizes . We denote the length of as and the number of substitutions as .
For the induction base we will prove that the lemma holds for length and an arbitrary number of substitutions . And we will also show that the lemma holds for every word length if there is only one substitution in the sequence of positive words which realizes . After that we are prepared to do the induction step.
For the induction step we fix arbitrary with . We then assume that the lemma holds for length if the number of substitutions in the sequence is . And we assume that if the length of is that it holds for every number of substitutions . We then show that it also holds for length and substitutions.
An Illustration of the structure of the double induction is depicted in Figure 9.
For the first part of the induction base let and let the realizing sequence contain an arbitrary number of substitutions. Since , both and consist of only one formal generator of , which must come from a reflection in . Thus, we have for reflections in .
Now, since , we already have , because any relation that could be applied on the left side is of the form for . But this is only true for and . Thus, a choice of and a positive word is possible such that the theorem is true for word length .
Now we want to show that the theorem holds for arbitrary length of when there is only one substitution in the sequence of positive words which realizes . In this case, we have