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Automaticity, almost convexity and falsification by fellow traveler properties of some finitely presented groups.

Elder, Murray J (2000) Automaticity, almost convexity and falsification by fellow traveler properties of some finitely presented groups. PhD thesis, The University of Melbourne.

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    Abstract

    We set out to examine the automaticity and almost convexity of an intriguing class of groups. Brady and Bridson provide examples from this class with quadratic isoperimetric function that are not biautomatic. Thus showing these examples are automatic would answer a long-standing question in automatic group theory. Wise gives another example from this class which is non-Hop�an and CAT(0). Determining the automaticity of this example would answer one of two questions; are all CAT(0) groups automatic, and are all automatic groups Hop�an? Determining its almost convexity would give similar insight. We start by trying to understand the geodesic structure of the Cayley graphs of these examples, for a particular choice of generating set. This leads us to de�ne the notion of a pattern in the Cayley graph, and we succeed in characterising the set of all patterns for these groups. From this we can prove they are almost convex for the chosen generating sets. This gives the fi�rst example of a non-Hop�an almost convex group. We also prove that the full language of geodesics is not regular, and moreover there is no geodesic automatic language for these examples with respect to the chosen generating sets. Neumann and Shapiro defi�ne the falsi�cation by fellow traveler property and show that if a group enjoys this property then its full language of geodesics is regular. Consequently the above examples do not enjoy this property. Related to it is the loop falsi�cation by fellow traveler property which we introduce in this thesis. Figure 2 summarises some facts about these properties. The two non-implications shown result from this thesis. We ask whether all groups with a quadratic isoperimetric function enjoy the loop falsi�cation by fellow traveler property. If so we would have a surprising characterisation for these groups. An example of Stallings appears to provide some clues to this question. We also examine the question of higher dimensional fi�niteness and higher dimensional isoperimetric functions for groups enjoying these geometric properties. We prove that if a group enjoys the falsi�cation by fellow traveler property then it is of type F3. We ask whether the larger class of almost convex groups are of type F3. Stallings' group would be a potential counterexample to this, since it is �finitely presented and not of type F3. We prove that for two independently arising generating sets, Stallings' group is not almost convex, suggesting it is not almost convex for any generating set.

    Item Type: Thesis (PhD)
    Subjects: 20-xx Group theory and generalizations
    Faculty: UNSPECIFIED
    Depositing User: Dr Murray Elder
    Date Deposited: 17 Sep 2012 10:41
    Last Modified: 17 Sep 2012 10:41
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/1211

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