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Topological methods in group theory : the adjunction problem
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Short, Hamish Buchanan (1983) Topological methods in group theory : the adjunction problem. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3921970
Abstract
The work presented here is a new method of attack on an old group theory problem known as the Adjunction Problem, defined by B H Neumann in 19^3 (see [Nj ). The problem is the following : given a group, form a new group by adding one new generator and one new relation ; determine the conditions under which the natural map from the original to the modified group is an injection. (For instance the new relator must not be conjugate to a word in the original group.) The main result obtained using the new methods is that the map is indeed an injection when the original group is locally indicable - a new result independently obtained by Howie [HJ and Brodskii [Brj .
Chapter 1 consists of some basic definitions and some of the known results, together with statements of the new results and some instances of where the problem arises in lowdimensional topology.
In Chapter 2 we introduce the new methods - showing that a non-trivial element in the kernal of the natural map for a given group and added relator (a "counter-example") gives us a labelled, planar graph with certain properties (a "special diagram") and that this special diagram in its turn defines a counter-example (these results are summed up in 2.22). These topologically obtained diagrams turn out (2.10) to be dual to the "Dehn diagrams" of Small Cancellation Theory (see for instance [Ls] or l_L2J ).
In Chapter 3 a class of such diagrams is constructed and it is shown that none of these corresponds to a counterexample. This class contains the only diagrams known to the author which give potential counter-examples ("triples") such that the new generator appears with exponent-sum non-zero in the added relator.
Chapter *+ begins with the construction of a potential function on a diagram, based on work by Lyndon [L2J . This is then used to prove the main result of the thesis, the Freiheits- satz for locally indicable groups, a new proof of the result which (as noted above) has been independently obtained by Howie and by Brodskii. Finally it is show that the existence of a counter-example for a given group G and a given relator r depends upon the existence of a counter-example for G* <s> and added relator r" , where r" is one of two words obtained from r using a homomorphism from G-*<t> to 71* which takes r to zero.
| Item Type: | Thesis (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Group theory, Topology, Adjunction theory | ||||
| Official Date: | November 1983 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Supervisor(s)/Advisor: | Rourke, C. P. (Colin Patrick), 1943- | ||||
| Sponsors: | Science Research Council (Great Britain) | ||||
| Format of File: | |||||
| Extent: | v, 70 pages : illustrations | ||||
| Language: | eng |
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