Simmonds, William Francis, 1963- (1991) Complete parameterized presentations and almost convex Cayley graphs. PhD thesis, University of Warwick.

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Abstract

This thesis is meant as a contribution to the theory of three classes of groups, those classes being the groups defined by complete parameterized presentations, automatic groups, and groups with almost convex Cayley graphs.

Chapter 1 is basically definitions and terminology. Chapter 2 is a short exposition of the theory of automatic groups; we prove only one major result in this chapter (due to (CHEPT)), i.e., that the abelian groups are automatic.

In chapter 3 we study presentations of groups and monoids which are complete (with respect to certain orderings of the words in their generators). Such presentations define monoids with fast solutions to their word problems. We define a class of (possibly infinite) presentations which we call r-porameterized, or of type Pr; these presentations are the central theme of this thesis. With the help of the computer program described in chapter 4, we demonstrate that there are group presentations which have infinite r-parameterized completions (i.e. complete supersets), but which have no finite completion with respect to any ShortLex ordering. The 1-parameterized presentations are, arguably, the simplest non finite presentations we can define (at least as far as groups are concerned), but we prove that completeness of such presentations is not in general decidable.

Chapter 4 is the description of a (short) program which attempts to complete 1-parameterized group presentations by the Knuth-Bendix method. We conclude the chapter with a short report on its implementation.

In chapter 5 we study groups with almost convex Cayley graphs. Such graphs are recursive, but the property of being almost convex does tend to be hard to prove or disprove in practice. We prove that the word length preserving complete groups and the least length bounded automatic groups have almost convex Cayley graphs. We believe that these are strict subclasses because (we shall prove) the group U(3,Z) is almost convex, but is already known not to be automatic and, we conjecture, it has no r-parameterized complete (ShortLex) presentation. We conclude chapter 5 with a slightly generalized, arguably simpler, algebraic proof of J.W. Cannon's theorem that the abelian by finite groups are almost convex.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Class groups (Mathematics), Semigroups, Infinite groups, Cayley graphs
Official Date:	November 1991
Dates:	Date Event November 1991 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Holt, Derek F.
Sponsors:	Science and Engineering Research Council (Great Britain)
Format of File:	pdf
Extent:	[5], 175 leaves
Language:	eng

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