Modeling rooted in-trees by finite -groups
In section 1 we describe the abstract graph theoretic foundations for a kind of infinite rooted in-trees with root , weighted vertices , and weighted directed edges . The vertex degrees are always finite but the trees contain infinite paths . In section 2 we introduce a group theoretic model of the rooted in-trees . Vertices are represented by isomorphism classes of finite -groups , for a fixed prime number . Directed edges are represented by epimorphisms of finite -groups with characteristic kernels . The weight of a vertex is realized by its nuclear rank and the weight of a directed edge is realized by its step size . These invariants are essential for understanding the phenomenon of multifurcation. Since the structure of our rooted in-trees is rather complex, we use pattern recognition methods for finding finite subgraphs which repeat indefinitely. Several periodicities admit the reduction of the complete infinite graph to finite patterns. Additionally, we employ independent component analysis for obtaining a graph dissection into pruned subtrees. As a coronation of this chapter, we show in section 3 that fork topologies provide a convenient description of very complex navigation paths through the trees, arising from repeated multifurcations, which are of the greatest importance for recent progress in determining -class field towers of algebraic number fields.
Key words and phrases:Rooted directed in-trees, descendant trees, infinite paths, vertex distance, weighted edges, pattern recognition methods, pattern classification, independent component analysis, graph dissection; finite -groups, limits, periodicity, extensions, nuclear rank, multifurcation, presentations, commutators, central series
2000 Mathematics Subject Classification:Primary 05C05, 05C07, 05C12, 05C22, 05C25, 05C38, 05C63, 05C70; secondary 20D15, 20E18, 20E22, 20F05, 20F12, 20F14
1. Underlying abstract graph theory
Let be a graph with set of vertices and set of edges . We expressly admit infinite sets and , but we assume that the in- and out-degree of each vertex is finite.
1.1. Directed edges and paths
In this chapter, we shall be concerned with directed graphs (digraphs) whose edges are rather ordered pairs than only subsets with two elements. Such a directed edge is also denoted by an arrow with starting vertex and ending vertex . Thus, we have . Now, infinitude comes in.
(Finite and infinite paths.)
A finite path of length in is a finite sequence of vertices such that for . We call , resp. , the starting vertex, resp. ending vertex, of the path. The degenerate case of a single vertex is called a point path of length .
An infinite path in is an infinite sequence of vertices such that for all . In this case, is the ending vertex of the path, and there is no starting vertex.
1.2. Rooted in-trees with parent operator
Our attention will even be restricted to rooted in-trees , that is, connected digraphs without cycles such that the root vertex has out-degree whereas any other vertex has out-degree . A vertex with in-degree at least is called capable whereas a vertex with in-degree is called a leaf. For a rooted in-tree we can define the parent operator as follows.
Let be a rooted in-tree. Then the mapping , , where is the unique edge with starting vertex , is called the parent operator of . For each vertex , there exists a unique finite root path from to the root ,
expressed by iterated applications of the parent operator, and with some length . Each vertex in the root path of is called an ancestor of .
The descendant tree of a vertex is the subtree of consisting of vertices with ancestor , that is , and edges .
A vertex is called an immediate descendant (or child) of a vertex , if there exists a directed edge . In this case, is necessarily the parent of .
We can define a partial order on the vertices of the tree by putting if , that is, if is descendant of , and is ancestor of . The root is the minimum.
The root is always a common ancestor of two vertices . By the fork of and we understand their biggest common ancestor, denoted by , which admits a measure.
(Vertex distance.) The sum of the path lengths from two vertices to their fork is called the distance of the vertices.
1.3. Mainlines and multifurcation
We shall also need weight functions with non-negative integer values for vertices , and with positive integer values for edges . In particular, the sets of vertices and edges have disjoint partitions
such that is precisely the set of leaves of the tree . Thus, there arise weighted measures.
(Path weight and weighted distance.)
By the path weight of a finite path with length in such that for we understand the sum . The sum of the path weights from two vertices to their fork is called the weighted distance of the vertices.
(Mainlines and minimal trees.) An infinite path in with edges of weight , that is, such that for all , is called a mainline in .
The minimal tree of a vertex is the subtree of the descendant tree consisting of vertices , whose root path in possesses edges of weight only, that is , and edges .
(Branches.) Let be a mainline in . For , the difference set of minimal trees is called the branch with root of the minimal tree . The branches give rise to a disjoint partition .
Finally, we complete our abstract graph theoretic language by considering arbitrary weights.
Let be a positive integer. A vertex has an -fold multifurcation if its in-degree is an -fold sum due to incoming edges of weight , for each . That is, we define counters of all incoming edges of weight , and additionally, we have counters of all incoming edges of weight with capable starting vertex,
We also define an ordering and a notation  for immediate descendants of by writing for the th immediate descendant with edge of weight , where and .
2. Concrete model in -group theory
Now we introduce a group theoretic model of the rooted in-trees in § 1. Vertices are represented by isomorphism classes of finite -groups , for a fixed prime number . Directed edges are represented by epimorphisms of finite -groups with characteristic kernels , where denotes the nilpotency class of and is the lower central series of .
We emphasize that the symbol is used now intentionally for two distinct mappings, the abstract parent operator , , in Definition 1.2, and the concrete natural projection onto the quotient , , for each individual vertex , which should precisely be denoted by , but we omit the subscript, since there is no danger of misinterpretation. In both views, is the parent of .
The weight of a vertex is realized by its nuclear rank [11, § 14, eqn. (28), p. 178] and the weight of a directed edge is realized by its step size [11, § 17, eqn. (33), p. 179]. These invariants are essential for understanding the phenomenon of multifurcation in Definition 1.7. In particular, we can hide multifurcation by restricting all edges to step size , that is, by considering the minimal tree instead of the entire descendant tree of a vertex . In our concrete -group theoretic model, all vertices of a minimal tree share a common coclass, which is the additive complement of the (nilpotency) class with respect to the logarithmic order of . Generally, the logarithmic order of an immediate descendant with parent increases by the step size, , since . Consequently, the coclass remains fixed in a minimal tree with , since
A minimal tree which contains a unique infinite mainline is called a coclass tree. It is denoted by when its root is of coclass . For further details, see [11, § 5, p. 164].
In view of the principal goals of this chapter, we must specify our intended situation even more concretely. We put , the smallest odd prime number, and we select as the root either or , characterized by its SmallGroup identifier . These are metabelian -groups of order , logarithmic order , class , and coclass .
2.1. Periodicity of finite patterns
Within the frame of the above-mentioned model with for the theory of rooted in-trees as developed in § 1, the following finiteness and periodicity statement becomes provable.
The virtual periodicity of depth-pruned branches of coclass trees has been proven rigorously with analytic methods (using zeta functions and cone integrals) by du Sautoy  in , and with algebraic methods (using cohomology groups) by Eick and Leedham-Green  in . We recall that a coclass tree contains a unique infinite path of edges with uniform step size , the so-called mainline. Pattern recognition and pattern classification concerns the branches.
(A finite periodically repeating pattern.)
Among the vertices of any mainline in , there exists a periodic root with and a period length such that the branches
are isomorphic finite graphs, for all . Up to a finite pre-periodic component, the minimal tree consists of periodically repeating copies of the finite pattern .
2.2. Graph dissection by independent component analysis
Dissection by Galois action
Figure 1 visualizes a graph dissection of the tree by independent component analysis. This technique drastically reduces the complexity of visual representations and avoids overlaps of dense subgraphs. The left hand scale gives the order of groups whose isomorphism classes are represented by vertices of the graph. The mainline skeleton (black) connects branches of non- groups (red) in the left subfigure and branches of -groups (green) in the right subfigure. This terminology has its origin in the action of the Galois group on the abelianization , when a vertex of is realized as second -class group of an algebraic number field . For quadratic fields , we obtain -groups.
A -group admits an automorphism acting as inversion on the commutator quotient .
The actual graph consists of the overlay (superposition) of both subfigures in Figure 1. Infinite mainlines are indicated by arrows. The periodic bifurcations form an infinite path with edges of alternating step sizes and , according to Theorem 2.2. We call it the maintrunk.
With the aid of Figure 1, a particular instance of Theorem 2.1 can be expressed in a more concrete and ostensive way by taking the tree root as the ending vertex of the mainline , and by using the variable class and the fixed coclass as parameters describing all mainline vertices . The periodic root is with and the period length is . The finite periodic pattern consists of the two branches (red) and (green). The pre-period is irregular and consists of the two branches (red) and (green). But is not coclass-settled, has nuclear rank , and gives rise to a bifurcation with immediate descendants (green) of step size .
Dissection by Artin transfers
In Figure 1, we have tacitly used a second technique of graph dissection by independent component analysis. Figure 2 is restricted to the coclass tree with exemplary root , which is the leftmost coclass tree in both subfigures of Figure 1. However, now this coclass tree is drawn completely up to logarithmic order , containing both, non- branches and -branches. The tree is embedded in a kind of coordinate system having the transfer kernel type (TKT) as its horizontal axis and the first component of the transfer target type (TTT) as its vertical axis [14, Dfn. 4.2, p. 27]. It is convenient to employ a second graph dissection, according to three fundamental types of transfer kernels:
the vertices with simple types , , and , , which are leaves (left of the mainline), except those of order ,
the vertices with scaffold (or skeleton) type , , which are either infinitely capable mainline vertices or non-metabelian leaves (immediately right of the mainline),
the vertices with complex type , , which are capable at depth and give rise to a complicated brushwood of various descendants (right of the mainline).
The tacit omission in Figure 1 concerns all vertices with complex type and the leaves with scaffold type. Our main results in this chapter will shed complete light on all mainline vertices and the vertices with simple types. Underlined boldface integers in Figure 2 indicate the minimal discriminants of (real and imaginary) quadratic fields whose second -class group realizes the vertex surrounded by the adjacent oval. Three leaves of type are drawn with red color, because they will be referred to in Theorem 3.1 on -class towers.
2.3. Periodicity of infinite patterns
With the aid of a combination of top down and bottom up techniques, we are now going to provide evidence of a new kind of periodic bifurcations in pruned descendant trees which contain a unique infinite path of edges with strictly alternating step sizes and , the so-called maintrunk. It is very important that the trees are pruned in the sense explained at the end of the preceding section § 2.2.2, for otherwise the maintrunk will not be unique. In fact, each of our pruned descendant trees is a countable disjoint union of pruned coclass trees , , which are isomorphic as infinite graphs and connected by edges of weight , and finite batches , , of sporadic vertices outside of coclass trees. The top down and bottom up techniques are implemented simultaneously in two recursive Algorithms 2.1 and 2.2.
The first Algorithm 2.1 recursively constructs the mainline vertices , with class , of the coclass tree , for an assigned value , by means of the bottom up technique. In each recursion step, the top down technique is used for constructing the class- quotient of an infinite limit group . Finally, the isomorphism is proved.
(An infinite periodically repeating pattern.) Let be an upper bound. An infinite path is generated recursively, since for each , the immediate descendant of step size of the second mainline vertex of the coclass tree , is root of a new coclass tree . The pruned coclass trees
are isomorphic infinite graphs, for each . Note that the nuclear rank .
This is the first main theorem of the present chapter. The proof will be conducted with the aid of an infinite limit group , due to M. F. Newman. Certain quotients of give precisely the mainline vertices with and , as will be shown in Theorem 2.3, Remark 2.2.
Theorem 2.2 remains true for any upper bound .
2.4. Mainlines of the pruned descendant tree
The complete theory of the mainlines in is based on the group
For each , quotients of are defined by
For each , and for each , quotients of are defined by
The following Algorithm 2.1 is based on iterated applications of the -group generation algorithm by Newman  and O’Brien . It starts with the root , given by its compact presentation, and constructs an initial section of the unique infinite maintrunk with strictly alternating step sizes and in the pruned descendant tree . In each step, the required selection of the child with appropriate transfer kernel type (TKT) is achieved with the aid of our own subroutine IsAdmissible(), which is an elaborate version of [13, § 4.1, p. 76]. After reaching an assigned coclass rhb, our algorithm navigates along the mainline of the coclass tree and tests each vertex for isomorphism to the corresponding quotient of class c2r1vb.
Input: prime p, compact presentation cp of the root, bounds hb,vb, sign s.
Code: uses the subroutine IsAdmissible(). r := 2; // initial coclass Root := PCGroup(cp); for i in [1..hb] do // bottom up in double steps along the maintrunk Des := Descendants(Root,NilpotencyClass(Root)+1: StepSizes:=); for j in [1..Des] do if IsAdmissible(Des[j],p,0) then Root := Des[j]; end if; end for; r := r + 1; // coclass recursion Des := Descendants(Root,NilpotencyClass(Root)+1: StepSizes:=); for j in [1..Des] do if IsAdmissible(Des[j],p,0) then Root := Des[j]; end if; end for; end for; c := 2r - 1; // starting class c in dependence on the coclass r er := pr; l := (c - 1) div 2; ec := pl; M<a,t> := Group<a,t|(at)p=ap,((t,a),t)=a(sp),aer=1,(t,a)ec=1>; QM,pM := pQuotient(M,p,c); // top down construction if IsIsomorphic(Root,QM) then // identification printf "Isomorphism for cc=o, cl=o.n",r,c; end if; for i in [1..vb] do // bottom up in single steps along a mainline c := c + 1; // nilpotency class recursion if (0 eq c mod 2) then // even nilpotency class l := c div 2; ec := pl; M<a,t> := Group<a,t|(at)p=ap,((t,a),t)=a(sp),aer=1,tec=1>; else // odd nilpotency class l := (c - 1) div 2; ec := pl; M<a,t> := Group<a,t|(at)p=ap,((t,a),t)=a(sp),aer=1,(t,a)ec=1>; end if; QM,pM := pQuotient(M,p,c); // top down construction Des := Descendants(Root,NilpotencyClass(Root)+1: StepSizes:=); for j in [1..Des] do if IsAdmissible(Des[j],p,0) then Root := Des[j]; end if; end for; if IsIsomorphic(Root,QM) then // identification printf "Isomorphism for cc=o, cl=o.n",r,c; end if; end for; Output: coclass r and class c in each case of an isomorphism.
Algorithm 2.1 is designed to be called with input parameters the prime p3 and cp the compact presentation of either the root with sign s-1 or the root with sign s+1. In the current version V2.22-7 of the computational algebra system MAGMA , the bounds are restricted to rhb8 and cvb2r135, since otherwise the maximal possible internal word length of relators in MAGMA is surpassed. Close to these limits, the required random access memory increases to a considerable value of approximately GB RAM.
(Mainline vertices as quotients of the limit group .) Let , .
For each , and for each , the mainline vertex of coclass and nilpotency class in the tree is isomorphic to .
For each , the projective limit of the mainline with vertices of coclass in the tree is isomorphic to .
is an infinite non-nilpotent profinite limit group.
(1) The repeated execution of Algorithm
for successive values from hb:=0 to hb:=6,
with input data
p:=3, cp:=CompactPresentation(SmallGroup(243,i)), ,
, and vb:=32,
proves the isomorphisms for and .
The algorithm is initialized by the starting group of coclass r:=2.
The first loop moves along the maintrunk recursively with strictly alternating step sizes and
until the root of the coclass tree with r=2+hb is reached.
The second loop iterates through the mainline vertices , , of the coclass tree
always checking for isomorphism to the appropriate quotient .
The subroutine IsAdmissible() tests the transfer kernel type of all descendants
and selects the unique capable descendant with type resp. .
(2) Since periodicity sets in for , the claim is a consequence of Theorem 2.1. (3) The quotient is already infinite and non-nilpotent. Adding the relation suffices to give central and profinite. ∎
Theorem 2.3 remains true for arbitrary upper bounds , .
When the top down constructions in Algorithm 2.1 are cancelled, the bottom up operations are still able to establish much bigger initial sections of the infinite maintrunk and of the infinite coclass tree with fixed coclass . Admitting an increasing amount of CPU time, we can easily reach astronomic values of the coclass, , and the nilpotency class, , that is a logarithmic order of , without surpassing any internal limitations of MAGMA, and the required storage capacity remains quite modest, i.e., clearly below GB RAM. This remarkable stability underpins Conjecture 2.2 with additional support from the bottom up point of view.
2.5. Covers of metabelian -groups
Only one of the coclass subtrees , , of the entire rooted in-tree contains metabelian vertices, namely the first subtree . The following theorem shows how transfer kernel types are distributed among metabelian vertices of depth on the tree , as partially illustrated by the Figures 1 and 2.
(Metabelian vertices of the coclass tree .)
For each finite -group , we denote by the nilpotency class, by the coclass, and by the transfer kernel type of . More explicitly, such a group is also denoted by . The following statements describe the structure of the metabelian skeleton of the coclass tree with root , resp. , down to depth .
For each , the mainline vertex of the coclass tree possesses type , , resp. , .
For each , there exists a unique child of with type , , resp. , .
For even , there exists a unique child of with type , , resp. , . Thus,