Chapters authored
Modeling Rooted in‐Trees by Finite p‐Groups
Graph theoretic foundations for a kind of infinite rooted in-trees T(R)=(V,E) with root R, weighted vertices v ∈ V, and weighted directed edges e∈E⊂V×V are described. Vertex degrees deg(v) are always finite but the trees contain infinite paths (vi)i≥0. A concrete group theoretic model of the rooted in-trees T(R) is introduced by representing vertices by isomorphism classes of finite p-groups G, for a fixed prime p, and directed edges by epimorphisms π: G → πG of finite p-groups with characteristic kernels ker(π). The weight of a vertex G is realized by its nuclear rank n(G) and the weight of a directed edge π is realized by its step size s(π)=logp(#ker(π)). These invariants are essential for understanding the phenomenon of multifurcation. Pattern recognition methods are used for finding finite subgraphs which repeat indefinitely. Several periodicities admit the reduction of the complete infinite graph to finite patterns. The proof is based on infinite limit groups and successive group extensions. It is underpinned by several explicit algorithms. As a final application, it is shown that fork topologies, arising from repeated multifurcations, provide a convenient description of complex navigation paths through the trees, which are of the greatest importance for recent progress in determining p-class field towers of algebraic number fields.
Part of the book: Graph Theory