Ph.D Thesis

Ph.D StudentWeiss Uri
SubjectFrom Combinatorial Plane Geometry to Word and Conjugacy
DepartmentDepartment of Mathematics
Supervisor ASSOCIATE PROFESSOR Arie Juhasz


Solving and understanding Max Dehn's word and conjugacy problems from 1911 remain to this day an active research subject. These problems where originally formulated regarding group presentation but can be also formulated for monoids (although there are some fine points regarding the conjugacy problem for monoids). Due to P. S. Novikov it is known that these problems have no general solution. Nevertheless, in many restricted but important cases solutions for the problems do exist.

One example where the word and conjugacy problems have a solution is when a group acts `nicely' on a sufficiently understood geometric space. This is the main theme of Geometric Group Theory. Actions of groups on geometric spaces are abundant. A prime example comes from group presentations which induce a natural action of the presented group on the associated Cayley complex (whose 1-skeleton is the Caylay graph). This action, which we regard as the internal geometry of the presentation, is usually not nice enough for the methods of Geometric Group Theory. However, the predecessor of this theory, namely, the so-called Combinatorial Group Theory, has a useful tool which allows understanding of the internal geometry; this tool is called van Kampen diagrams. In this work we solve the word and conjugacy problems in three different setups by understanding the structure of van Kampen diagrams (and thus some of the internal geometry of the problem at hand).

One way the power of our approach can be seen is from its application to monoids, for which the usual tools of Geometric Group Theory do not work as well. Besides monoids, we apply the method in other two non-standard setups. In the first case, the geometry of the van Kampen diagrams is complicated and thus are handled using derived diagrams. In the second case, we start from information of the structure of the van Kampen diagrams and derive the solution to Dehn's problems.