Conway-Fried-Parker-Voelklein type results on braid orbits/Hurwitz spaces

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 (called the B(ranch)G(eneration)-Thm below) is a contribution to that topic. Modulo (*²), it describes precisely the Hurwitz space components and their definition fields. To understand this discussion you need to know the following things (see Nielsen-Classes.html):

What is a Nielsen class (defined by G and C) and a little about equivalences on them.
How braids act on elements of a Nielsen class.
How Hurwitz space components interpret as braid orbits in this action.

Here are our topics:
I. Motivation from the regular version of the Inverse Galois Problem (RIGP):
II. Practical Lifting Invariant discussion:
III. B(ranch)-G(eneration)-Theorem (CFPV-Thm):
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], has its Main Theorem on the case where r' is 1, and C' is the 3-cycle class in A_n. This is is a good starting point for understanding the definitions. This also shows, in practice the CFPV-Thm is a start for many main stream applications. This paper also discusses why modular curves, and their definition fields are special cases of these considerations.

The problems raised in [twoorbit] and [hurwitzLiu-Oss.pdf] by the pure-cycle case – especially how this case led to using cusp types to understand the Main Conjecture of Modular Towers – are tractible. They point to explicitness results that go beyond that high C'-support hypothesis in the CFPV-Thm. Pure-cycles are merely conjugacy classes in S_n represented by elements with just one disjoint cycle of length exceeding 1.

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 our discussion, assume G has no center. It also simplifies things to assume characteristic 0 fields are our main concern. We denote the rational numbers by Q.

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)ⁱⁿattached to the Nielsen class, and that means:

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

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

Example: In the case of pure-cycle Nielsen classes (in, say A_n) 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 A_n.

The gist of the [inv_gal, Main Thm.] 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)ⁱⁿhas 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. That starts 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 SM_G of G.

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

A sub-example is n=5, so A_n=PSL₂(5) (2x2 matrices in Z/5 of determinant 1, mod ± I₅) and Spin_n=SL₂(5). As in [hf-can0611591] when C → C_3^r (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,…,g_r) in O₊(resp. O_-) – satisfying the product-one condition g₁^…g_r =1 – to 2' elements g'=(g'₁,…,g'_r) in Spin_n. Then, you get g'₁^… g'_r ∈{±I_n}. This value, the lifting invariant, is a braid orbit invariant for which ±I_n 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 |SM_G| 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:

Elements g ∈ C'₁ have lift to R having the same order as g; or
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 generated by commutators in SM_G 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/p^t instead of Z/p if the Schur multiplier is not an elementary abelian group.)

Example: Assume n ≥ 4 and g ∈ A_n of order 2 is a product of 2s disjoint cycles. Then, any lift of g to the Spin_n has order 4 if s is odd, and order 2 if s is even [h4-0104289, Prop. 5.10]. The Schur multiplier of S_n (not a 2-perfect group) has order 4. 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 S_n Nielsen class braid orbits.)

II.b. Computing the lifting Invariant: Consider again the case G=A_n 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 g_i'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,… ,d_r}=d, the genus of such cover is g_d, as in the formula: 2(n+g_d-1)= ∑_i=1^r (d_i-1). When g_d=0 (the assumption for all Liu-Osserman cases), the lifting invariant is (-1)^(d_i2-1)/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. B(ranch)-G(eneration)-Theorem (CFPV-Thm): Assume G centerless and C' a distinct rational union of (nontrivial) classes in G. An infinite set I_G,C' indexes distinct absolutely irreducible Q varieties Θ_G,C' = Θ_G,C',Q ={H_i}_{i∈I_G,C'}. This collection satisfies:

There is a finite-one map i∈I_G,C'→_iC (consisting of r_i unordered conjugacy classes of G supported in C' ); and
the RIGP holds for G with conjugacy classes C supported in C' ⇔ there is i∈I_G,C'with H_i having a Q point.

IV. More on the RIGP and what from CFPV-Thm gives the BG-Thm: The emphasis above is on I_G,C' being infinite. Regular realizations of G come by augmenting existence of Θ_G,C' with info on the varieties {H_i}. 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: The following proven what being explicitness about the indexing sets I_G,C'means.

V.a.When G=S_n, C' has one class, that of a 2-cycle: Then the indexing set is just the number of 2-cycles, r ≥ 2(n -1) (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 S_n 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=D_p^k+1 with p odd, and C₂ (conjugacy class of an involution), then i → C_2^r is one-one and onto, with r=2i ≥ 4. Also, H_i^rd (the reduced version of the Hurwitz spaces) identifies with the space of cyclic p^k+1 covers of hyperelliptic jacobians of genus (2i-2)/2 [NonRigidGT,§5].

If G=A_n with C' =C₃, class of a 3-cycle, then i → C_3^r with r ≥ n is two-one. Denote indices mapping to r by i^±. Those covers in H_i^± are the Galois closures of degree n covers X→ P¹_z with i 3-cycles for local monodromy. For r=n-1 the map i → C_3^r 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 S_n (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. This showed that from any p-perfect group with a nontrivial Schur multiplier, you could produce examples of Nielsen classes for which there was 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, who found a way to put more structure into the appearance of the Schur multiplier, clever though totally ineffective. Serre asked me about the 3-cycle case (when the covers have genus 0; r=n-1 in the A_n 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 because I wanted to prove results on two precise applications. 1st: To the Inverse Galois Problem and Alternating group presentations of G_Q, 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 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.

[#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 A_n, 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).