Small cancellation in lacunary hyperbolic groups, and the word and conjugacy problems

Lacunary hyperbolic groups were introduced by Olshanskii, Osin and Sapir in 2009.
One can think about this class of groups as a natural generalization of hyperbolic groups from the class of finitely presented groups to the class of finitely generated groups. This class of groups is known for containing some infinite “monster” groups such as torsion free Tarski monsters, verbally complete groups, etc. I will discuss lacunary hyperbolic groups, obtained by applying special small cancellation conditions, such that their word and conjugacy problems have properties that were not known before.