Word and conjugacy problems, and Tarski monsters

The word and conjugacy problems for groups were introduced by Max Dehn in 1911 and are widely recognized as some of the most important features of finitely generated groups. In the talk we will discuss Tarski monsters (i.e. finitely generated non-cyclic groups with (finite) cyclic proper subgroups) with effective word and conjugacy problems. We will also underline the method of such constructions and discuss some if its applications in a wider context.