Abstract

Decision problems in group theory have been a topic of much interest for some time. The standard formulation for such problems goes along the lines of “Given a finite group presentation P, does there exist an algorithm to determine some property of the group described by P?” For many such questions, the answer is no. However, in certain cases, if the collection of groups being considered is restricted to satisfying some condition (say, being abelian, hyperbolic, etc), then many of these decision problems can be answered. In this talk I will give examples of such decision problems that are undecidable in general, but can be decided when we impose further conditions. In addition to this, I will outline other conditional decision problems whose decidability is (to the best of my knowledge), still unknown. The most interesting such example is the following (open) question: Given a finite presentation of a non-trivial group, can one algorithmically construct a non-trivial element?