Candidate: Paula T. Smith

Degree of: Doctor of Philosophy

Department: Mathematics

Title: Local Symmetries of Symmetrical Cayley Maps

Committee:
Dr. John Martino, Chair
Dr. Joseph Buckley
Dr. Terrell Hodge
Dr. Arthur White
Dr. Robert Jajcay

Date: Friday, April 26, 2002 4:00 p.m. - 6:00 p.m.
Alavi Commons Room, 6th Floor, Everett Tower

Abstract:
Graphs, groups, and surfaces are all subjects of study in topological graph theory, using techniques and principles from the disciplines of graph theory, algebra, and topology. A Cayley graph provides a graphical representation of a finite group and a fixed generating set for the group; a Cayley map is a two?cell imbedding into a surface of a Cayley graph such that labeled outward?directed darts occur in the same sequence at each vertex. A dart is a directed edge. In this work, we generalize Cayley maps to allow two?cell imbeddings of graphs with loops and multiple edges.

The set of all automorphisms of a Cayley map M form a group, Aut M. A Cayley map that achieves its maximum possible number of automorphisms is called a symmetrical Cayley map. An automorphism of a Cayley map permutes the dart labels. This action induces a homomorphism Psi: Aut M ?? > S_k, where S_k is the symmetric group on the dart labels. The image of Psi is the automorphism group of a symmetrical Cayley map N (see Chapter 2). The original Cayley map M is a covering space over N. Hence M and N have the same local structure.

Let rho be a cyclic permutation denoting the common sequence of dart labels at each vertex of M. A symmetrical Cayley map is called balanced if, for all x, rho(x^{?1})=[rho(x)]^{?1}, and antibalanced if rho(x^{1})=[rho^{?1}(x)]^{?1}. For a balanced map the outward?directed darts in a neighborhood of a vertex are either all labeled by involutions or if not, then a labeled dart and its inverse are symmetric across the vertex, while for antibalanced maps labeled darts and their inverses are reflected across a common line in the tangent plane of the vertex.

A t?balanced map is a generalization of these ideas, i.e., for all x, rho(x^{?1})=[rho^t(x)]^{?1}. This concept may be further generalized by the introduction of a superscript function f, i.e., for all x, rho(x^{?1})=[rho^{f(x)}(x)]^{?1}. A symmetrical Cayley map is balanced if f is constantly 1, antibalanced if f is constantly ?1, and t?balanced if f is constantly t. The homomorphism Psi can be used to identify when a symmetrical Cayley map M is t?balanced. Let k be the degree of each vertex in M. If Psi(Aut M)=Z_k, then M is balanced, if Psi(Aut M)=D_k (the dihedral group of order 2k), then M is antibalanced, and if Psi(Aut M) is a more general semidirect product mathbb Z_k rtimes mathbb Z_2, then M is t?balanced, the value of t being determined by the action of mathbb Z_2 on mathbb Z_k [Richter?Siran?Jajcay?Tucker?Watkins, Martino?Schultz?Skoviera].

In this Theses, we describe the distribution of inverses around a vertex for a t?balanced map. We also describe the distribution of inverses for the case where the superscript function alternates between two values a and b. In this case, the inverse distribution is a blend of two ?balanced distributions. The image of Psi corresponding to the case of the alternating superscript function is also determined. It is an extension of a dihedral group by a cyclic group. Furthermore, when k/2 is odd the extension is a semidirect product, and when a+b/2=1 k/2, then Im Psi is a wreath product Z_{k/2} wr Z_2.

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