Descriptions of Groups using Formal Language Theory

This work treats word problems of finitely generated groups and variations thereof, such as word problems of pairs of groups and irreducible word problems of groups. These problems can be seen as formal languages on the generators of the group and as such they can be members of certain well-known language classes, such as the class of regular, one-counter, context-free, recursively enumerable or recursive languages, or less well known ones such as the class of terminal Petri net languages. We investigate what effect the class of these various problems has on the algebraic structure of the relevant group or groups. We first generalize some results on pairs of groups, which were previously proven for context-free pairs of groups only. We then proceed to look at irreducible word problems, where our main contribution is the fact that a group for which all irreducible word problems are recursively enumerable must necessarily have solvable word problem. We then investigate groups for which membership of the irreducible word problem in the class of recursively enumerable languages is not independent of generating set. Lastly, we prove that groups whose word problem is a terminal Petri net language are exactly the virtually abelian groups.