Nielsen Classes

Two well-known objects both define a Nielsen class:
  1. A compact Riemann surface X with a nonconstant analytic function f.
  2. An algebraic relation g(z,w)=0 between two independent variables z and w (both appearing in the equation). We then consider z as the first of the variables.
Both #1 and #2 define a cover of compact algebraic curves where the target curve is the projective line P1z.  In fact, both are equivalent to having such a cover.  We understood P1z to be the Riemann sphere from 1st year complex variables. That would be the complex plane C∪∞, or for an algebraist it would be K∪∞ for the projective line over any field K.

The file Alg-Equations.html has a direct discussion of the equivalence of #1 and #2. Since books like Springer's and Gunning's Riemann Surfaces are over 50% dedicated to a substantially self-contained proof of this equivalence, there are serious questions about the nature of this equivalence, especially to what discipline the proof belongs.

Hermann Weyl's much earlier book was a source for both Springer and Gunning. Weyl's book did mathematics an astounding favor by resurrecting Riemann's work – once the uniformization principle had a solid proof.  The latter assured that compact Riemann surfaces had one of three spaces – the sphere, the plane or the disk – as their universal cover. The file Alg-Equations.html  discusses this more. Here we accept what came from those developments, and we get to what Nielsen classes have to do with algebraic relations. The first two topics below are on this page. The last four are in NielsenClassesCont.html.
  1. What is a Nielsen class?
  2. Equivalences on Nielsen classes vs equivalences on covers
  3. Families of covers and the introduction of braids
  4. Interpreting Hurwitz space components as braid orbits on Nielsen classes
  5. Examples from dihedral groups and pure-cycles
  6. Why do we need (the complication of) Nielsen classes?
    7.
The topics under I are the following:
I.1. The precise definition and relations between Nielsen classes:
I.2. Some FAQs on the Nielsen Class conditions:

The topics under II are the following:
II.1. The pair (G,C) attached to a cover:
II.2. The Nielsen class attached to a cover:
II.3. Absolute equivalence of covers:
II.4. Inner equivalence of covers:
II.5. Reduced and other equivalence of covers:

I. What is a Nielsen class?: A group G with a set of r conjugacy classes C={C1,…,Cr} in G defines a Nielsen class. Its elements consist of r-tuples in G, g=(g1, …, gr) ∈Gr, satisfying three extra conditions. 

For any g ∈ G, denote the conjugacy class of g by Cg. We use the notation g ∈C to mean: In some order the collection   {Cgi}i=1r is exactly the same as C, including the multiplicity with which each class appears.

I.1. The precise definition and relations between Nielsen classes: Here is the definition of Nielsen class:

Ni(G,C)•={g ∈C∩(G)r |  g1g2…gr=1 (product-one) and <g>=G (generation)}•.

The • is a place for decoration indicating an equivalence relation on the set. The choice of equivalence in applications is extremely important. There is always a non-trivial equivalence – necessarily represented by a group action – when Nielsen class elements represent P1z covers.

To get more from Nielsen classes, we use collections of them that relate different groups through their conjugacy classes. Here is notation for some of these. Each naturally allows comparing their attached Hurwitz spaces.

Sometimes, when G ≤ G', we extend Cg to be a conjugacy class, CgG', in G'. That is convenient notation for many uses of the B(ranch) C(ycle) L(emma). That is our main tool figuring the definition field of Hurwitz spaces, and often of their components when they have more than one.

At other times G might be the (homomorphic) image of a group G*, with kernel ker(G*→ G). In general, several conjugacy classes in G* map to Cg.  Modular Towers, however, uses the case when ker(G*→ G) has order prime to the order of g and a unique conjugacy class in G* lies over Cg.

Then, we either rely on the context to see Cg means class in G*, or we use the notation g ∈ C ∩(G*)r. In this way, the homomorphism G*→ G induces Ni(G*,C)• → Ni(G,C)•. Variants of this appear throughout the results because such maps on the Hurwitz spaces give a handle on Hurwitz space properties not available classically.

A major source of results uses the relation between Nielsen classes defined by different equivalences, especially the map from inner to absolute equivalence from §II.3 and §II.4. See Nielsen-ClassesCont.html for the uses of that map and examples that show the behavior that guides many results.

I.2. Some FAQs on the Nielsen Class conditions:

Must G be a finite group?: No, that is not necessary! Yet, that is the case that defines a Hurwitz space. We go beyond that, for example, in studying Modular Towers. There it is natural, starting from a finite group, to consider various profinite covers of it. 

Can we switch the order of the gis in the product-one condition?: No! The only time you can safely do that is when G is abelian. We have often heard researchers insist they were doing the abelian case first; later they would get to the general case. Except, later never seems to happen. Hurwitz-Spaces.html starts with a section called “The Abelian Case is Not a Good Model.”

We have kept the order on entries of g inviolate in the product-one condition. Yet, for most applications not on the appearance of the conjugacy classes. Yes, many constructions use a temporary order on the conjugacy classes, and then as a separate step, unorder them for applications. Modular curves has a special case of this.

An ordering there on conjugacy classes interprets as an ordering on the branch points of a Weierstrass model of an elliptic curve. Indeed, it is standard to take one point at ∞. Yet, in most applications the remaining branch points are unordered. The result of ordering would be a different (reduced Hurwitz) space than the traditional X0(p).

Can't you just use cycle-types rather than conjugacy classes?: No! That suffices for only a few applications. Yet, it can be valuable to use cycle-types to help handle Nielsen classes. If a Nielsen class deals with degree n sphere covers, then G has an attached degree n permutation representation. That is, an embedding G ≤ Sn.

That allows considering the conjugacy classes CSn in Sn. Still, if G is not Sn, then Ni(G,CSn ) doesn't make sense, but Ni(Sn,CSn ). An example: G is the alternating group An, with n=5. We have two 5-cycles g1=(1 2 3 4 5) and g2=(1 3 2 4 5). Then, Cg1 is distinct from Cg2, but Cg1S5=Cg2S5: (2 3) conjugates the one 5-cycle to the other, but no element in  A5 does. In this case, no matter how often we repeat the conjugacy classes of 5-cycles in C, Ni(Sn,CSn ) will be empty, even though Ni(An, C) may not be. §V and §VI in Nielsen-ClassesCont.html say more on this.

Isn't it hard to track more than one condition at a time? Which is most important?: This is an excellent question. Even the conjugacy class condition – when you are not in Sn – is non-trivial. Then, there are the product-one and generation conditions. It can be hard to even list, efficiently, elements in a Nielsen class. So, here is a guide for which pure-cycle Nielsen classes are helpful. It often pays to handle the generation condition first. Then consider the geometrically significant product-one condition.

II. Equivalences on Nielsen classes vs equivalences on covers: The word equivalent used in the introduction forces a subtle topic, the different types of equivalences on covers, and therefore on Nielsen classes.

II.1. The pair (G,C) attached to a cover: Given a cover f: X → P1z , of degree n, we associate to it a Nielsen class. Things simplify if we assume X is a connected – one component. We don't need the equivalence with condition #2 to form the Galois closure group of f.

Alg-Equations.html explains the minimal Galois closure ^f: ^X → P1z. It is the minimal cover factoring through f having as many automorphisms factoring through P1z as its degree. It is a connected component of the n-fold fiber product of the map f. It is transparent that Sn acts on the n-fold fiber product as permutations. The Galois group Gf of the connected component is the subgroup of Sn preserving component. Then, Gf=G.

For each integer d' let ζd' be e2πi/d'. There are a finite number of special – branch – points on P1z: where the fiber of f has fewer than n points. Denote these {z1, …, zr}=z. Take a disk Di on P1z around each zi. Then, the restriction of f over f –1(Di) consists of covers of Di, ramified at exactly one point, zi. There may be several components. Let fD': D' → Di be restriction to one of these. Assume its degree is d'.

Alg-Equations.html (or in detail ) explains (and points to proofs at this site) for the rest of this subsection and the next. 1st year complex variables shows the following: D' is a disk, and there is an analytic change of variables so that fD' becomes w' → w'd'. Further, restriction of some element gzi ∈Gf to D' (regarded as a disk around the origin) induces the automorphism w → ζd'w. While gzi is not unique, its conjugacy class Ci in Gf is.

II.2. The Nielsen class attached to a cover: Finally, it is possible to select gi ∈Ci so that g=(g1,…,gr) ∈Ni(G,C). This depends on selecting a special collection – a classical set of generators  (or a topological bouquet) – of fundamental group generators for r-punctured sphere Uz = P1z \ {z}. 

This gives R(iemann)'s E(xistence) T(heorem) a dependence on topology (see Alg-Equations.html). Existence of g – an algebraic statement, even if its proof is not – is the supreme fact attesting that f  is in the Nielsen class. So, we comment further. Given a bouquet P ={P1,…,Pr} of paths on Uz based at z0∈Uz, all elements in the Nielsen class correspond to covers constructed from P.

This is how we say the construction P → gP: It maps a bouquet to a branch cycle description of the cover. The only ambiguity is how we named the cover fiber over z0. Renaming the fiber then amounts to conjugating gP by a permutation.

That appears to allow any permutation from Sn, and so G could be conjugated by any element in Sn. Yet, as the point of Nielsen classes is to be able to compute, that makes unnecessary complications. Instead, we fix G and restrict to permutations in the normalizer, NSn(G), of G in Sn. This works because we can often compute this normalizer easily.

Conversely, given elements in the Nielsen class you can construct the covers: A Nielsen class produces a cover having the Nielsen class as its branch cycle description.

Less obvious: You need just one element in each (nonempty) Nielsen class to get them all. This trick works by going to a covering Nielsen class where all Nielsen class elements are in the outer automorphism orbit of G.

Yet, being precise about corresponding Nielsen classes and covers requires being precise about corresponding equivalences respective equivalences on the two sets. That is our next topic.

Note: if you change P, then you change that correspondence, but not the Nielsen class. That is, ambiguity in the choice of P is the source of the braid group action (see III. in Nielsen-ClassesCont.html). Also, the process called “Grabbing a cover by its Branch Points” in §V of Hurwitz-Spaces.html preserves the Nielsen class of the cover. So, given one cover in Nielsen class, we can find another with its branch points whereever we want them.

II.3. Absolute equivalence of covers: There are several variants of Hurwitz spaces. The strongest equivalence on covers in a Nielsen class that preserves the data (G,C) is called absolute.  Some use the terminology of mere covers to refer to the equivalence class.

Given two covers f1: X1 → P1z and  f2: X2 → P1z , they are absolute equivalent (in the same mere cover class) if there is ψ: X1→ X2 (continuous is sufficient) commuting with the maps f1 and f2. This forces f1 and f2 to have the same branch points. So we can use the same bouquet P to consider what are the relations between elements in the Nielsen class.

The correspondence above means the branch cycle descriptions g1 and g2 for f1 and f2 relate by g1= hg2h-1 with h ∈NSn(G) conjugating each entry of g2. That means absolute equivalence of covers corresponds to a Nielsen class where • means to mod out by NSn(G) on Ni(G,C): Ni(G,C)abs =  Ni(G,C)/NSn(G). 

II.4. Inner equivalence of covers: Consider the Galois closure construction from §II.1 on a cover  f: X → P1z. A choice was made: We picked a connected component of the n-fold fiber product of f. Another choice would have given an isomorphic copy of G, but not necessarily the same. Again we see the normalizer NSn(G).

Suppose, however, we insist in our equivalence between two covers, say  f1 and f2, in § II.3, that we also stipulate this choice of component. Then, the only ambiguity in giving an isomorphism of a fixed copy of G and the Galois closure group of f1 is an inner isomorphism of G: conjugation by an element of G.

Note: If G has a (non-trivial) center, then several elements of G induce the same inner isomorphism of G. That is, there may be different maps between the Galois closures of f1 and f2 that induce the same inner isomorphism of the Galois Groups.

So, centers cause us some problems. On the other hand, in Modular Towers there is no way to avoid them, and we handle many cases of center quite well. Just not the case where an abelian group is a quotient of G. For this much more ancient methods work.

Conclusion: Inner equivalence on covers stipulates that the cover come with a component of its n-fold fiber product and the equivalence between them extends to a map between their respective components. Thankfully this gives a simple equivalence on Nielsen classes. That is, • is just modding out by G: Ni(G,C)in =  Ni(G,C)/G.

II.5. Reduced and other equivalences of covers: Mobius transformations, elements of PGL2 act as the automorphism group of P1z. Thus, α ∈ PGL2 equivalences f: X → P1z and αof: X → P1z. Since that moves the branch  points,  it is an equivalence that gets added non-trivially to any other equivalence. The page Nielsen-ClassesCont.html finishes this equivalence by explaining how it interprets on Nielsen classes. It only changes the Nielsen class equivalence for r=4.

There are many equivalences we haven't considered here. We briefly mention some from the beginning of the subject. They remain helpful in analyzing Hurwitz spaces, and/or detecting the relation between other classical moduli spaces and Hurwitz spaces.
  1. Replace • by modding out by any group properly between G and NSn(G).
  2. Add markings to points of  f: X → P1z lying over branch points (equivalence preserves the markings).
  3. Equivalence f1: X1 → P1z and  f2: X2 → P1z if their vector bundles over P1z  are equivalent.
  4. Various combinations of the equivalences above.
[#1, §4] allowed the most general construction of equivalences on spaces of covers. Item #b was the topic of [#2], giving one approach to identifying the modular curve X(n) with a Hurwitz space (there are others). The conclusions of [§6, #3] on the equivalence of the families of Davenport covers are an example of #c.

[#1] Fields of definition of function fields and Hurwitz families and groups as Galois groups, Communications in Algebra 5 (1977), 17–82.
[#2] with R. Biggers, Moduli Spaces of covers of P1and Representations of the Hurwitz monodromy group, J. für reine und angew. Math. 335 (1982), 87–121.
[#3] Relating two genus 0 problems of John Thompson, Volume for John Thompson's 70th birthday, in Progress in Galois Theory, H. Voelklein and T. Shaska editors 2005 Springer Science, 51–85.

Pierre Debes and Mike Fried, 04/16/08