Proper decompositions of finitely presented groups
For a finitely presented group G, determining whether it admits a proper decomposition as an amalgamated free product or HNN-extension is often not straightforward. In a presentation 2-complex for G, there are special subsets called patterns and tracks. Bass-Serre theory and van Kampen's theorem show that given such a G, there exists a proper decomposition if and only if there exists a track that determines a proper decomposition. With this in mind, an original algorithm is constructed here to find a basis set of patterns, such that every pattern is a unique rational linear combination of these basis patterns. Another desirable pattern to determine for such a G is a maximal set of disjoint tracks (or MSD). Here we construct an algorithm to generate an MSD for a finitely presented group G, or terminate early providing a track that gives a proper decomposition of G. Part of this algorithm includes the proof that if a group, not immediately obviously admitting a proper decomposition, has a proper decomposition as an HNN-extension, then the algorithm will find a track providing such a proper decomposition. The concepts of basis patterns and MSDs were developed after the counter-example given here to the main conjecture of A.N. Bartholomew's Proper Decompositions Of Finitely Presented Groups (1987), that the sum of two tracks giving trivial decompositions gives a trivial decomposition itself, was discovered. A proof is given using Bass-Serre theory on a certain class of groups that if every track in an MSD determines certain trivial decompositions of G, then G has no proper splitting over a finite subgroup. One of the main aims of this research was to provide a computer program dealing with patterns. This has been achieved with the program Presentation accompanying this thesis. It enables easy manipulation and graphical display of patterns, and allows many operations to be performed on them. Also both algorithms given in the thesis are implemented. It is hoped that the theory of patterns will be advanced by this new accessibilty.