In 1955, Adian proved a remarkable theorem showing that if $\mathcal{P}$ is a Markov property (most non-trivial group properties are Markov; for example: being hyperbolic, being finite and being abelian are all Markov properties) of finitely presented groups, then $\mathcal{P}$ is not recursively recognizable. In other words, given a finite group presentation $G=\langle S|R \rangle$, it is impossible to algorithmically decide whether $G$ has property $\mathcal{P}$ or not.

I will define Markov properties, discuss what “algorithmically decidable” means and show how to prove the Adyan-Rabin theorem starting from a group with undecidable word problem. If time permits, I will give a brief outline of how to construct a group with unsolvable word problem.