On Dehn's algorithm

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46211/1/208_2005_Article_BF01361168.pd

The SQ universality of some small cancellation groups

PhDA group G is a small cancellation group if, roughly, it has a presentation G= <A; R with the property that for any pair r, s of elemets of R either r=s1 or there is very little free cancellation in forming the product rs. The classical example of such a group is the fundamental group of a closed orientable 2-manifold of genus k which has a presentation k G=< al, bl, ..., ak, bk; 'TT \ai, bi' i=1 A countable group G is SQ-universal if every countable group can be embedded =in some quotient of G. The obvious example of SQ-universal group is the free group of rank 0. This work is a study of the SQ-universality of some small cancellation groups. A theory of diagrams is investigated in some detail- to be used as a tool in this study. The main achievement in this work is the following two results: (1) With few exceptions a small cancellation group contains nonabelian free subgroups. ( The emphasis here is on the nature of the free generators. ) (2) A characterization of the S Q-universality of some small cancellation groups

Creating Characters in a Story-Telling Universe

Extended story generation, i.e., the creation of continuing serials, presents difficult and interesting problems for Artificial Intelligence. We present here the first phase or the development of a program, UNIVERSE, that will ultimately tell extended stories. In particular, alter describing our overall model of story telling, we present a method for creating universes of characters appropriate for extended story generation. This method concentrates on the need to keep story-telling universes consistent and coherent. We also describe the information that must be maintained for characters and interpersonal relationships, and the use of stereotypical information about people to help motivate trait values. The use of historical events for motivation is also described. Finally, we present an example of a character generated by UNIVERSE

Solvability of the Conjugacy Problem for Hnn Extensigns of Finitely Generated Free Abelian Groups

This study is concerned with the solvability of the conjugacy problem for HNN extensions of finitely generated free abelian groups. In order to prove the theorem, normal forms and a solution of the word problem for such groups are required. In Chapter II we show that the word problem and generalized word problem for a finitely generated abelian group are solvable. Then the normal form for HNN extension of finitely generated abelian groups immediately follows, and by Corollary 2.1 the word problem is solvable for these groups. In Chapter III the conjugacy problem for long words is shown to be solvable. To clarify the theorem an example is given at the end of Chapter III. To complete the algorithm we show that solving the conjugacy problem for short words of the elements of these groups is equivalent to a certain decision problem for polynomials with complex rational coefficients.Mathematic

A computation of the action of the mapping class group on isotopy classes of curves and arcs in surfaces

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982.MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCEBibliography: leaves 155-156.by Robert Clack Penner.Ph.D

String rewriting systems and associated finiteness conditions

We begin with an introduction which describes the thesis in detail, and then a preliminary chapter in which we discuss rewriting systems, associated complexes and finiteness conditions. In particular, we recall the graph of derivations r and the 2- complex V associated to any rewriting system, and the related geometric finiteness conditions F DT and F HT. In §1.4 we give basic definitions and results about finite complete rewriting systems, that is, rewriting systems which rewrite any word in a finite number of steps to its normal form, the unique irreducible word in its congruence class. The main work of the thesis begins in Chapter 2 with some discussion of rewriting systems for groups which are confluent on the congruence class containing the empty word. In §2.1 we characterize groups admitting finite A-complete rewriting systems as those with a A-Dehn presentation, and in §2.2 we give some examples of finite rewriting systems for groups which are A-complete but not complete. For the remainder of the thesis, we study how the properties of finite complete rewriting systems which are discussed in the first chapter are mirrored in higher dimensions. In Chapter 3 we extend the 2-complex V to form a new 3-complex VP, and in Chapter 4 we define new finiteness conditions F DT2 and F HT2 based on the homotopy and homology of this complex. In §4.4 we show that if a monoid admits a finite complete rewriting system, then it is of type F DT2 • The final chapter contains a discussion of alternative ways to define such higher dimensional finiteness conditions. This leads to the introduction, in §5.2, of a variant of the Guba-Sapir homotopy reduction system which can be associated to any co~­ plete rewriting system. This is a rewriting system operating on paths in r, and is complete in the sense that it rewrites paths in a finite number of steps to a unique "normal form"

Topological methods in group theory : the adjunction problem

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* and added relator r" , where r" is one of two words obtained from r using a homomorphism from G-* to 71* which takes r to zero