Nielsen Classes Continued

This file is an introduction to more advanced topics on Nielsen classes. We continue the section numbering of the file Nielsen-Classes.html. §II in that file introduced a number of equivalences placed on Nielsen classes. There is a simple idea behind these equivalences.

For a given Nielsen class, any cover in that Nielsen class – up to some natural equivalence of covers – should correspond to a point on a parameter space: A point versus an equivalence class of covers. Covers of the sphere come from Riemann's Existence Theorem via covering space theory applied to fundamental group of the r-punctured Riemann sphere. You apply the theory by having the fundamental group act as permutations on some set.

So, it would be great if such parameter spaces came from covering space theory applied to some space that isn't so far from the Riemann sphere. Well, they do! It is the fundamental group of the projective r-space "punctured" (minus) its discriminant locus, acting as permutations on a Nielsen class. That is the gist of what this file introduces.

We do much more pointing at papers in this file to keep down the technicalities. Still, there is one extremely valuable starting point: A Nielsen is a generalization of the genus of a Riemann surface. Therefore these parameter spaces are generalizations of the space of Riemann surfaces of genus g. How we use these spaces to get theorems is, of course, the deepest part of the results. For that we mostly point to the papers.

The Riemann-Hurwitz Formula: Here, again, is the definition of Nielsen class:

Ni(G,C)•={g ∈C∩(G)r |  g1g2…gr=1 (product-one) and <g>=G (generation)}• where the • is to indicate what equivalence is in force, and there is a permutation representation of G putting it as a subgroup of Sn for some integer n. Any element g of Sn has orbits on the set of integers {1,…,n}. We count orbits (including those of length 1), and subtract that from n. The result is the index ind(g) of g. This indication of orbits of g is the cycle type of g.

For example, g=(1 2 3) ∈ S4 has index 4 - 2 = 2. Notice, if we regarded g as in S5, the calculation would give the same index, but in the form 5 - 3 = 2. Typically we assume – that is necessary to get the genus formula right – that G is a transitive subgroup of Sn.

R-H: Given a cover f: X → P1z in Ni(G,C)•, take any Nielsen class representative g Then, the genus geX of X appears in the following formula:  2(n + geX -1) = Σi=1r ind(gi).

Example: Suppose your Nielsen class consisted of 4-tuples order 2 elements in the dihedral group of order 2p, in its standard representation of degree p. Each such element has (p-1)/2 disjoint cycles of order 2, and there are four of them. So the right side of the genus formula gives 2(p-1), while the left side gives 2(p - geX -1). Conclude the genus of such a cover is 0.
  1. Families of covers and the introduction of braids
  2. Interpreting Hurwitz space components as braid orbits on Nielsen classes
  3. Examples from dihedral groups and pure-cycles
  4. Why do we need (the complication of) Nielsen classes?
III. Families of covers and the introduction of braids:

IV. Interpreting Hurwitz space components as braid orbits on Nielsen classes:

V. Examples from dihedral groups and pure-cycles:

VI. Why do we need (the complication of) Nielsen classes?:

Notice the computation of the genus of a cover in a Nielsen class (see R-H) came just from knowing the cycle types of the entries of a Nielsen class representative. The cycle type of a set of conjugacy classes in a group G inside Sn is obtained by a forgetful functor: You regard the conjugacy classes as in Sn. I'm going to give just one example of why conjugacy classes versus cycle types matter. Notice, it came up in the first serious applications of the braid-monodromy method.

So, it is a surrogate for all other serious uses, even though it was about covers given by polynomial maps. That is, the resolution of this problem required the full force of the theory – in a special case – because you don't find polynomial maps trivially, except in the case of the general polynomial of degree n. The original paper is [Fr73], though a fuller, more expository treatment [Fr10] takes advantage of all the developments since then.

Notice the big deal is the use of the very general B(ranch)C(ycle)L(emma), even though the appearance of the simple group classification and the braid group seem more powerful. For example, the resolution of Davenport's problem over Q required no classification.

Finally, since this is the end of the elementary html offerings on Nielsen classes, I also mention the paper [Fr95] which I wrote expressly  show how important was the idea of conjugacy classes in very elementary situations. The situation there was just degree n polynomials with An as group. The paper shows how to use the B(ranch) C(ycle) L(emma) and the monodromy method to disprove two well-known conjectures about polynomial maps (covers). The author of the article's Math Review – John Swallow – calls it a service to the community. The html file explains the two problems. It is also a primer on handling nontrivial points on explicit families – of polynomial covers – in arithmetic geometry. From this we see exactly when the conjectures do hold.

Sometimes one must make conjectures, for to gain attention to a projectrequiring several years to work, requires focussing the attention ofothers on how your work will impinge on theirs. For me, the fewconjectures I've put in public have been about Modular Towers () wherethe ideas here get worked in more daring situations. Mostly, however, Iwork privately on conjectures, but rarely – anymore – callsattention to them in print.

VI.1. Introduction to Davenport's problems and some reductions:

Davenport's problem was essentially to classify polynomials over anumber field by their ranges on almost all residue class fields. Thatis, given two polynomials fand h withcoefficients in a number field K,suppose theirranges (as sets) are equal on almost all residue class fields. We callthese a Davenport pair over K.

Theproblem was to find out if they must be linearly related: f(x)= h(ax+b) – the trivialcase – with aand bconstants. Usually a  andb – if theyexist – must automatically end up in K. (Always underour indecomposability assumption below.) Davenport asked just over Q,where the result was substantially different than it is over generalnumber fields.

Since much has now been written on this, I emphasize just theconclusions that get us thinking about Nielsen classes asbeing related to conjugacy classes, not just cycle types. Theresults impinge directly on Riemann's Existence Theorem, on grouptheory and use of the classification of finite simple groups, and onnumber theory. To get such information also required an extra conditionon the polynomials. Indeed, there is much reason to continueinvestigations on Davenport's Problem today. I'll explain why below.

VI.2. Indecomposability and results about the monodromy groups ofcorresponding polynomial covers:

Recall Möbius equivalence for rational functions: The Möbiusclass of fconsists of the rational functions obtained from f by composiing onthe inside and outside by a linear fractional transformation. This isdifferent than being linearly related, because we included composing onthe outside.

A polynomial fis indecomposableover K ifand only if it is not a composition h1(h2(x))of two lower degree polynomials. We denote by Tfthe natural degree npermutation representation of either the geometric (Gf: The Galois closure group of the cover over an algebraic closure) orarithmetic (^Gf: the Galois closure group of the cover over K) monodromy groupof f.Denote by Gf(1) the elements in Gf that stabilize 1.

Key Observations early in the investigations [Fr70, Thm. 1]:
  1. A polynomial f(of degree n)has among its branch cycles an n-cycleσ∞ at ∞. It has just one, unless the geometricmonodromy group Gfis cyclic, and this is equivalent to f  beingin the Möbius class of the cyclic polynomial xn. 
  2. If fis decomposable over the algebraic closure, then it is decomposableover K.
  3.  fis decomposable if and only if Tfis a primitive  representation:There is no group properly between Gf(1) and Gf.
  4. If Tfprimitive, then Tfis doubly transitiveunless f is (Möbius equivalent to) a Chebychev polynomial or acyclic polynomial. 
Doubly transitive is stronger than primitive, for it meansthat Gf(1) is transitive on {2,…,n}.So, once you join a single element that moves 1 then you have all of Gf.

VI.2. Monodromy groups of Davenport pairs:

The following are in [Fr73, ]and there is a slower exposition in[Fr10, ]. The result dependsheavily on that n-cycleat ∞.

Representation Thm: For (f,h) aDavenport pair:
  1.  deg(f) = deg(h),and the Galois closures of the covers of f and h are thesame, so Gf= Gh.
  2.  Also, Tf = Thas group representations, but not as permutation representations.
What [Fr73] is actually attacking is Schinzel's problem: When could f(x)-h(y), as a polynomial in separated variables, be reducible (factor as a polymial in two variables). It solves this under the assumption f is indecomposable. One result, under this assumption, is that with no loss, by replacing h by a composition factor, this problem is equivalent to Davenport's. Except Schinzel's problem has no number theory to it, while Davenport's sounds like it is only number theory.

At this point you can't see a Nielsen class, and you certainly can't see why conjugacy classes are neccessarily more significant to this problem, than is the cycle type. That happens in the next subsection.

VI.3. More than one conjugacy class of n cycles:

In [Fr73, Lem. 5] we learn a fact with a fancy name attached to it. The conditions #5 and #6 above show that we have a group which has a structure called a design – from the topic of projective geometry – and there is a non-multiplier in that design. That is actually saying that are several distinct conjugacy classes of n-cycles in Gf. Not just distinct, but there is no conjugation of one to the other by an element of Sn. But there is a  collineation taking one design to the other. Then, that somehow takes the one polynomial, f, to the other, h.

The existence of a non-multipier of the design translates to the proof of Davenport's statement in Da1 below. I got the idea of the proof of that from Tom Storer, who died in 2008. That prompted relooking at this topic. [Fr10, ] explains this. Then, it goes on to give a proof in Nielsen classes for that result by directly showing, essentially, that a Davenport pair of polynomials must be complex conjugate.

Further, [Fr73, p. 134], based on evidence from several chapters of [Ha63], conjectures that there is a list – projective linear groups over finite fields – that have a natural pair of permutation representations, that one can get one's hands on. Further, these and one other group, already contain all possibly monodromy groups of possible Davenport polynomials over any number field.

Then, there is one other step, a reason why from that infinite list you could possibly hope to produce a finite list of Nielsen classes that include all that correspond to actual Davenport pairs. That is, you could also "produce" them from this data. There were two methods here, once James Ax suggested I involve Walter Feit. Mine used Nielsen classes directly, and Feit's – using the data above – went directly after the character table of these projective linear groups. Both were dependent on using [R-H] and that polynomials give genus 0 covers.

Feit got his results into print in a journal on which he was an editor, quickly. Mine, which also included the exact description of the Nielsen classes, and using the braid group, the exact properties of these polynomials, mysteriously found problems getting into print, even as I corrected errors in Feit's version of the final list. Comments on that list are in the last subsection.

VI.4. Statement of results and the Genus 0 Problem:

Da1 Over Q two such polynomials with the same range are linearly equivalent: obtainable, one from the other, by a linear change of variables.

Da2 Actual Davenport pairs have a finite list of possible degrees, and they occur in families with very precise properties.

We attend to these general questions in [Fr10]. 
  1. What allows us to produce branch cycles, and what was their effect on the Genus 0 Problem (of Guralnick/Thompson)?
  2. What is in the kernel of the Chow motive map, and how much is it captured by using (algebraic) covers?
  3. What groups arise in 'nature' (a 'la a paper by R.Solomon)?
[Fe73] W. Feit, Automorphisms of symmetric balanced incomplete blockdesigns with doubly transitive automorphism groups, J. of Comb. Th. (A)14: (1973), 221–247.

[Fe80] W. Feit, Some consequences of theclassication of the finite simple groups, Proc. of Symp. in Pure Math.37 (1980), 175–181.

[Fr70]On a Conjecture ofSchur, Michigan Math. J. Volume 17, Issue1 (1970), 41–55 (pdf also on-line at theMichigan Math Journal). It gives the classification of exceptionalpolynomials – those that map one-one on infinitely many residue fields– of a number field. Schur's 1921 Conjecture generated much literature:at its solution Charles Wells sent me a bibliography of over 550papers, most showing certain families of polynomials – given by theform of their coefficients – contained none with the exceptionalityproperty. SchurConj70.pdf

[Fr73] M. Fried, Thefield of definition of function fields and a problem in thereducibility of polynomials in two variables, IllinoisJournal of Math. 17, (1973), 128–146. The pdf fileis a scan. The paper's center is the solution of Davenport'sProblem UMStory.html is a user-friendly guide to this paper and other problems it influencedthrough the monodromy method. It explains the firstserious use of a B(ranch)C(ycle)L(emma)for information on the defining field of an algebraicrelation.  dav-red.pdf

[Fr80] M. Fried, Expositionon an Arithmetic-Group TheoreticConnection via Riemanns Existence Theorem, Proceedings ofSymposia inPure Math: Santa Cruz Conference on Finite Groups, A.M.S. Publications37 (1980), 571–601.

[Fr95] Extension ofConstants, Rigidity, and the Chowla-Zassenhaus Conjecture,Finite Fields and their applications, Carlitz volume 1 (1995), 326–359:

[Fr99] M.D. Fried, Separatedvariablespolynomials and moduli spaces, Number Theory in Progress (Berlin-NewYork) (ed. J. Urbanowicz K. Gyory, H. Iwaniec, ed.), Walter de Gruyter,1999, Proceedings of the Schinzel Festschrift, Summer 1997:

[Fr10] VariablesSeparated Equations and Finite Simple Groups: This is a amore complete version of UMStoryShort.htmlwhose pdf file appears in the UMContinuum. Thatincluded two new tools: the B(ranch)C(ycle)L(emma)and the Hurwitz monodromy group. By walking throughDavenport's problem with hindsight, variables separated equations letus simplify lessons on using these tools. UMStory.pdf

[Ha63] M. Hall, The Theory ofGroups, MacMillan, NY 1963.