Conway-Fried-Parker-Volklein Theorem: Hurwitz space components/Braid Orbits on a Nielsen class

Results of C(onway)F(ried)P(arker)V(oelklein) start with any given finite group G and any collection of distinct (nonidentity) generating conjugacy classes C'={C'1,…,C'r'} of G. Occasional references appear to give hints where you can find the quoted results. We will refer to C' as the seed classes (and each C'i as a seed class). The CFPV subject is how to decipher components of a Hurwitz space defined by G and conjugacy classes C subject to:

(*) C is supported in the seed classes: It has C'-support.

An extra condition appears often below:

(*2) Each seed class appears with high multiplicity in C: It has high C'-support.

The CFPV-Thm (also called the B(ranch)-G(eneration)-Thm below) contributes to that topic. Modulo (*2), it describes precisely the Hurwitz space components and their definition fields. To understand this discussion you need these ideas (see Nielsen-Classes.html):
Here are our topics:
I. Motivation from the regular version of the Inverse Galois Problem (RIGP):
II. Practical Lifting Invariant discussion:
III. CFPV-Thm (B(ranch)-G(eneration)-Theorem):
IV. More on the RIGP and what from the CFPV-Thm gives the BG-Thm:
V. Explicit indexing sets:
VI. Comparing the 3-cycle and Liu-Osserman Pure-cycle case:
VII. Some history of CFPV-Thm:

The html file attached to hf-can0611591 explains the Main Theorem: The case where r' is 1, and C' is the 3-cycle class in An. This case will help you understand the definitions. This also shows, in practice the CFPV-Thm is a start for many main stream applications. Modular curves, and their definition fields, are special cases.

Pure-cycles are conjugacy classes in Sn represented by elements with just one disjoint cycle of length exceeding 1. twoorbit (building on hurwitzLiu-Oss.pdf ) uses the pure-cycle case to establish explicitly the Main Conjecture of Modular Towers using cusp types. This produces systems of spaces based on alternating groups that aren't modular curves, yet that exhibit modular curve-like properties. The idea: modular curves are based on dihedral groups, and other finite groups have their associated spaces figuring in properties of algebraic curves generalizing elliptic curve studies of modular curves. Understanding these spaces depends on identifying connected components of Hurwitz spaces, and therefore on going beyond the high C'-support hypothesis in the CFPV-Thm.

I. Motivation from the regular version of the Inverse Galois Problem (RIGP): The Inverse Galois Problem (Conjecture) asks for a finite group G how to realize G as the (Galois) group of an extension of each number field. To simplify this RIGP discussion, assume G has no center. Also assume that characteristic 0 fields are our main concern. Section CFPV.thm has two parts: The RIGP for G – a stronger statement – is that G is the Galois group of an extension L/Q(z) (z transcendental) with L having only Q as constants. According to the B(ranch)C(ycle)L(emma): Such a regular realization gives a cover, the cover defines a(n inner) Nielsen class ni(G,C)in and C is a rational union of conjugacy classes. Further, such a regular realization produces a Q point on the Hurwitz space H(G,C)in attached to the Nielsen class, and that means:

(*3) H(G,C)in must have an absolutely irreducible component (with its natural map to the space of r unordered branch points Ur)  over Q.

We denote the rational numbers by Q. [inv_gal, Main Thm.] gives the precise cyclotomic definition field of a Hurwitz space (either absolute or inner) defined by a specific Nielsen class (extending the absolute case given in [HurMonGG.pdf, Thm. 5.1]). Its geometric components, however, may have much larger definition fields. By adding the lift invariant and assuming conjugacy classes with large enough multiplicity, we can assure there are no surprising geometric components.

The rational union condition on C is equivalent to H(G,C)in having definition field Q.

Example: In the case of pure-cycle Nielsen classes (in, say An) when the cycles are n or n-1-cycles (depending on n odd or even) you are forced to include  both conjugacy classes that have this cycle type to get a rational union. Otherwise, for pure-cycle classes, rational union is automatic in An.

The gist of the [inv_gal, Cor. 1] is this:

(*4) Any G has an explicit finite cover, so that by replacing G with that cover, you can say there are infinitely many (distinct) such C so that each of the  H(G,C)in has such a Q component.

This suffices to prove cases of the generalization of Shafarevic's conjecture in [GQpresentation] and in the hands Thompson and Volklein (Völklein's book) to produce the first serious cases of higher rank Chevelley groups for which the RIGP holds. It is by being explicit about Hurwitz spaces that realizations have occurred. We now start our discussion of lifting invariants and the CFPV-Thm.

II. Practical Lifting Invariant discussion: I will happily explain what I understand on Schur multipliers. I here give examples: [rims-rev, §4] and [h4-0104289, §5.7] explain their applications.

II.a. Schur Multipliers: Toward the general result, consider first this special case. You have conjugacy classes in G which I label as C' (maybe just one). Special assumption:

(*5) All elements in C' have order prime to the order of the Schur multiplier SMG of G.

Example: When G=An, (*5)  is equivalent to all classes have odd order elements (n ≥ 4). Here the Schur multiplier is Z/2={±In} = ker(Spinn→ An), with Spinn the universal central extension of An.

A sub-example is n=5, so An=PSL2(5) (2x2 matrices in Z/5 of determinant 1, mod ± I5) and Spinn=SL2(5). As in [hf-can0611591] when C → C3r (r 3-cycles) and r ≥ n, then there are two braid orbits O+ and O- on the Nielsen class distinguished as follows.

Suppose you lift an r-tuple g=(g1,…,gr) in O+ (resp. O-) – satisfying the product-one condition g1…gr =1 – to 2' elements g'=(g'1,…,g'r) in Spinn. Then, you get g'1… g'r ∈{±In}. This value, the lifting invariant, is a braid orbit invariant for which ±In correspond respectively to the Nielsen class subsets O±.

Recall: A Frattini cover H→G of groups is a covering homomorphism in which no proper subgroup of H maps onto G. It is central if the kernel is in the center of H. Now I explain what happens if I drop (*5).

It comes down to the following. Instead of the whole Schur multiplier, consider just a prime p dividing |SMG| and a Z/p quotient of it. Such a quotient corresponds to a central Frattini cover R → G with ker(R → G)=Z/p. We now assume, among the C', there is a C'1, say, where p divides the order of its elements. Then, one of these can happen:
  1. Elements g ∈ C'1 have lift to R having the same order as g; or
  2. all their lifts to R have order p times the order of g.
Having conjugacy classes satisfying #2 in C' requires modding out by the ambiguity in lifting to R. Running over the conjugacy classes C' in C', mod out by the group AMG generated by commutators in SMG of the form g'h'(g')-1(h')-1=def(g',h') with g ∈ C', h ∈ G, and ' indicating lifts of both to R. (It suffices to consider one prime at a time, but you may need Z/pt instead of Z/p if the Schur multiplier is not an elementary abelian group.)

There is always central p Frattini extension Rp→G with kernel the whole p-part of the Schur multiplier of G. If G is p-perfect, then this extension is unique. The fiber product of these over G is the full representation cover (take this to be R) over G. Mod out R by the group AMG, to get RC' →G. Denote its kernel by MC'. We use this below.

Example: Assume n ≥ 4 and g ∈ An of order 2 is a product of 2s disjoint cycles. Then, any lift of g to the Spinn has order 4 if s is odd, and order 2 if s is even [h4-0104289, Prop. 5.10]. The Schur multiplier of Sn (not a 2-perfect group) is a copy of the Klein 4-group, Z/2 × Z/2. You can figure out what happens to conjugacy classes of order divisible by 2 in lifting to Z/2 quotients of the Schur multiplier from our previous example. (See Sn-SchurMult.html for the effect on Sn Nielsen class braid orbits.)

II.b. Computing the lifting Invariant: Consider again the case G=An and all conjugacy class elements have odd order. Then, you can – often quite practically – compute the value of the lifting invariant on a braid orbit in a Nielsen class. The method, based on the induction procedure of [hf-can0611591, §4.4], has one case based on the Fried-Serre formula for the case of genus 0. Since Nielsen classes are about compact Riemann surface covers of the sphere, there is an implicit permutation representation attached to a Nielsen class (made explicit in absolute or inner Nielsen classes). In the absolute alternating group case, when the permutation representation is the standard representation, each of the gi's is a product of disjoint cycles given as part of Nielsen class data. Any surface attached to such an r-tuple in the Nielsen class has an easily computed genus from the Riemann-Hurwitz formula.

In the simplest case, of pure-cycles of respective (odd) lengths {d1,… ,dr}=d, the genus of such cover is gd, as in the formula: 2(n+gd-1)= ∑i=1r (di-1). When gd=0 (the assumption for all Liu-Osserman cases), the lifting invariant is ∑i=1r (-1)(di2-1)/8.

Even in the pure-cycle case there is a more sophisticated application – though the formula has numerically the same complexity – in computing types of cusps on corresponding reduced Hurwitz spaces. The goal of this [,§4] is to divine when pure-cycle Nielsen classes define Modular Towers whose cusp structure looks like that of modular curves.

III. CFPV-Thm (B(ranch)-G(eneration)-Theorem): Use the notation of § II.a for the conjugacy classes C' and MC'. Here is the basic result:

Geometric (B(ranch)-G(eneration)-Theorem: For all Nielsen classes Ni(G, C), with C supported in C', where each conjugacy class in C' appearing with suitably high multiplicity, there are exactly |MC'| braid orbits, and therefore the same number of geometric components of the corresponding Hurwitz spaces.

The 2-cycle result and 3-cycle results have a common element. They proceed by showing that every Nielsen class element braids to an element close to a H(arbater)-M(umford) representative (g1,g1-1,g2,g1-1,…,gr,gr-1) for an appropriate value of r. For the 2-cycle you (easily) braid to an H-M rep. and for the 3-cycle case (dependent on which degree n we have), you can braid to one of an explicit list of tuples juxtaposed with an H-M rep.

Conway and Parker, given the data from Fried that combines the 3-cycle and 2-cycle results, introduced something like an H-M rep. that would work universally. Their major idea was to use their universal element to put a semigroup structure on the union of braid orbits on all the Nielsen classes Ni(G, C), with C running over all classes supported in C'. Following their rough outline, and correcting some points, [inv_gal, App.] does the case when C' includes all non-trivial conjugacy classes of G.

We discuss, however, below how far the result – partly because of the wasteful universal element. Further, it is non-trivial to compute |MC'|. Therefore [inv_gal, App.] produced covers of G for which we knew |MC'|=1.

Arithmetic (B(ranch)-G(eneration)-Theorem: Assume G centerless and C' a distinct rational union of (nontrivial) classes in G. An infinite set IG,C' indexes distinct absolutely irreducible Q varieties ΘG,C' = ΘG,C',Q ={Hi} i∈IG,C'. This collection satisfies:
  1. There is a finite-one map i∈IG,C' →iC (consisting of ri unordered conjugacy classes of G supported in C' ); and
  2. the RIGP holds for G with conjugacy classes C supported in  C'  ⇔ there is i∈IG,C' with Hi having a Q point.
The conjugacy class collections run over rational unions of conjugacy classes that also have high multiplicity of support in C'. Then, the component with lifting invariant 1 will certainly have definition field Q. The centerless condition is to assure fine moduli holds for the Hurwitz space components, and that therefore a Q corresponds to an actual regular Galois realization of G. Anyone desiring to use this result over finite fields needs only to know two things. IV. More on the RIGP and what from CFPV-Thm gives the BG-Thm: The emphasis above is on IG,C' being infinite. Regular realizations of G come by augmenting existence of ΘG,C' with info on the varieties {Hi}. To get this infiniteness, the Fried-Voelklein addendum says if you repeat each of the classes in C'  often enough (condition (*2)), then a value of a lifting invariant determines the corresponding irreducible component. This gives infinitely many such varieties, but it doesn't give the full list of all possible varieties where a Q point might be found because it requires all elements of C' appear often, and it is hard to figure just what often means. Also, if you care about K points, where K is a larger field, then the indexing sets in #3 and #4 will change. Still, modulo using the BCL, the result is qualitatively the same.

V. Explicit indexing sets:  What follows is a discussion on the indexing sets IG,C' .

V.a.When G=Sn, C' has one class, that of a 2-cycle: Then the indexing set is just the number of 2-cycles, r (even and ≥ 2(n -1) from Riemann-Hurwitz). Then, it is the oldest argument in the book that there is one orbit if this condition holds. Voelklein's book, Lem. 10.15, repeats my (essentially) 2-line argument. Since the Schur multiplier of Sn has order 4, it is surprising that this Schur multiplier does not split the Nielsen class into braid orbits. Sn-SchurMult.html explains why not.

V.b. Dihedral and Alternating group cases:  [hf-can0611591] discusses both. Here C' has one class. In the dihedral case it is that of the involution. If G=Dpk+1 with p odd, and C2 (conjugacy class of an involution), then i → C2r is one-one and onto, with r=2i ≥ 4. Also, Hird (the reduced version of the Hurwitz spaces)  identifies with the space of cyclic pk+1 covers of hyperelliptic jacobians of genus (2i-2)/2 [NonRigidGT,§5].

If G=An with C' =C3, class of a 3-cycle, then i → C3r with r ≥ n is two-one. Denote indices mapping to r by i±. Those covers in  Hi± are the Galois closures of degree n covers X→ P1z with i 3-cycles for local monodromy. For r=n-1 the map i → C3r is one-one.

VI. Comparing the 3-cycle and Liu-Osserman Pure-cycle case: A connectedness result of Liu and Osserman ([hurwitzLiu-Oss.pdf] while postdocs at Berkeley) says: if you take pure-cycle conjugacy classes in Sn (one disjoint cycle in the conjugacy class representatives), and the genus of covers in the Nielsen class is 0, then the absolute Hurwitz space is connected. The overlap with the result is the case r=n-1 for 3-cycles. These cases were especially interesting because whether the Main Modular Tower conjecture was true was 100% a story of analyzing my classification of cusps using the lifting invariant.  [#1, §6]  has a general guiding statement for the inverse Galois applications of these types of results – a la, Fried-Voeklein:  FS-Lift-Inv.html has  examples of pure-cycle Nielsen classes which show how the lifting invariant gives information on this potential umbrella result.

VII. Some history of CFPV-Thm: A first version of the B(ranch)C(ycle)L(emma) appeared in [#3, §3].  The version now used by many starts [#2, §5] (repeated in Matzat-Malle and in Völklein's book).  The Conway-Fried-Parker-Völklein result happened because I showed John Thompson the use of the Schur multiplier in 1988. From this you could use any group with a nontrivial Schur multiplier to produce infinitely many Nielsen classes having more than one braid orbit. I used 3-cycles – with a partial proof that the lifting invariant gave the precise components – as an application loaded case. John told Conway and Parker, as described above. asked me about the 3-cycle case (when the covers have genus 0; r=n-1 in the An cases above) from a talk I gave at Delong-Pisot-Poutieu seminar in Paris (1988). Serre based his proof [#4] – his 1st paper on the topic – on the braid method I used to show cases of this.

I waited a long time to send out for publication the paper hf-can0611591.html on the full result for all r so as to include two precise applications. 1st: To the Inverse Galois Problem and Alternating group presentations of GQ, the absolute Galois group of Q. 2nd: To giving useful “automorphic functions” (from theta nulls) on some of those components.  Both apply Riemann's subject of half-canonical classes. The last uses Serre's 2nd paper [#5] related to this topic.

Many seem to have trouble picking up the definition of Nielsen class, a generalization of the idea of conjugacy class. John Thompson has suggested the problem is because, to most mathematicians, all groups are symmetric groups. So, distinguishing conjugacy classes from disjoint cycle types is subtle them. Whatever the reason, the subject is far from ineffable, and Nielsen classes, and their equivalences, are the fundamental sets in understanding families of Riemann surface covers. The only geometric invariant so far introduced that separates components further, that might otherwise be indistinguishable, is the lifting invariant.

[#1] M. Fried, The Main Conjecture of Modular Towers and its higher rank generalization, in Groupes de Galois arithmetiques et differentiels (Luminy 2004; eds. D. Bertrand and P. Dèbes), Sem. et Congres, Vol. 13 (2006), 165–230.
[#2] M. Fried, Fields of definition of function fields and Hurwitz families and groups as Galois groups, Communications in Algebra 5 (1977), 17–82.
[#3] M. Fried, The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Illinois Journal of Math. 17, (1973), 128–146.
[#4] J.–P. Serre, Relêvements dans An, C.R. Acad. Sci. Paris 311 (1990), 477–482.
[#5] J.–P. Serre, Revêtements a ramification impaire et  thêta-caractèristiques,  C.R. Acad. Sci. Paris 311 (1990),  547–552.
[#6] H. Völklein, Groups as Galois groups, Cambridge Studies in Advanced Mathematics, vol. 53, Cambridge University Press, Cambridge, 1996.  MR1405612 (98b:12003).