Decision problems for finitely generated groups: overview and perspectives

I will present a historical account of the study of computability on finitely generated groups.
I will first survey some classical results that concern the study finitely presented groups.
We will then see that a recent theorem of Groves and Wilton introduces a completely new perspective on the subject: the ‘correct’ description of groups that need to be used to study decision problems was not the one that was studied. It is as though we had plenty of results about functions on the left computable reals and far fewer about functions defined on the computable reals.
I will discuss some research directions which follow from the Groves-Wilton theorem.