Title:

The application of pictures to decision problems and relative presentations

Regard the presentation P =< x; r > as a 2complex. Then we have the second homotopy module 2(P). The elements of 2(P) can be represented by spherical pictures. This is the key idea for this thesis. We give this preliminary background in Chapter 1. In Chapter 2, we study properties of groups concerning 2 (P). We show that all these properties are recursively unsolvable, that is, there are no effective methods which can be applied to an arbitrary finite presentation P to determine whether or not groups have these properties. Our main results are Theorems 2.3.1 and 2.3.2, that is, that pCockcroft and efficiency are recursively unsolvable. Let P be a collection of spherical pictures over P. Then we may form a 3complex K =< x, r; P >. In Chapter 3 we establish the picture problem for Kthe analogue of the word problem for P, a dimension higher. We prove Theorem 3.1.1.the existence of K with unsolvable picture problem. From now onwards, we deal with relative presentations. We are interested in investigating the asphericity of P =< H,t;th1th2th3t1h4 > (hi H). In Chapter 4, we survey the basic concepts, the important theorems for relative presentations and the tests for asphericity. The first major case that we consider is < H,t;t3at1b > where a and b are nontrivial elements of H. We investigate asphericity of this form in Chapter 5. Excluding some exceptions that are not yet decided, we state our results in Theorems 5.1.1 and 5.2.1. In Chapter 6, we consider the second major case < H,t;t2atbt1c > where a, b and c are nontrivial elements of H. As in Chapter 5, we have some exceptions and we state our results in Theorems 6.1.1, 6.2.1, 6.3.1 and 6.4.1.
