# Applications of Star Complexes in Group Theory

El-Mosalamy, Mohamed Soliman Hassan (1987) Applications of Star Complexes in Group Theory. PhD thesis, University of Glasgow.

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## Abstract

The main work of the thesis starts with Chapter 2. Chapter 2 concerns free subgroups of C(4). T(4) groups. Collins has investigated the free subgroups of groups with presentations satisfying the C(4). T(4) conditions. He has shown that such groups contains a free subgroup of rank 2 except in some cases which he lists explicitly. The exceptions are all two generator groups. In Chapter 2 we give a simple proof, using star complexes, that if G is a C(4). T(4) group and if G can not be generated by fewer than three elements then G contains a free subgroup of rank 2. (In fact we prove a slightly stronger result.) Much work has been done for C(4)-T(4). and C(6)-complexes. However T(6)-complexes have not so far been studied very much. For that reason our main work in the thesis is to study T(6)-complexes. This work is contained in Chapter 3 and Chapter 4. In Chapter 3 we give some examples of T( 6)-complexes and also examples of related complexes called hyperbolic complexes. In Chapter 4 we obtain new solutions to the word and conjugacy problems for T(6)-complex, and we discuss the dependence problems in general. ( The word problem is DP(1). and the conjugacy problem is DP(2).) In Chapter 5 we introduce the idea of the degree of a presentation, and also property-ST(m) and property-st(m). (These concepts are related to valences of vertices in star-complexes.) We ask whether having a presentation of degree m. property-STC m) or property-st(m) (m>2) puts any restriction on the group defined by the presentation. We show that the answer is no.

Item Type: Thesis (PhD) Doctoral Theoretical mathematics 1987 Enlighten Team glathesis:1987-77519 Copyright of this thesis is held by the author. 14 Jan 2020 11:53 14 Jan 2020 11:53 http://theses.gla.ac.uk/id/eprint/77519