Morse geodesics in lacunary hyperbolic groups

Elisabeth Fink (ENS, Paris, France)

Abstract: A geodesic is Morse if quasi-geodesics connecting points on it stay uniformly close. If the embedding of the cyclic subgroup generated by an element is a Morse geodesic, then that element is called a Morse element. In many known examples, Morse geodesics in groups have been found via Morse elements. By studying asymptotic cones and using small cancellation, we will show how Morse geodesics can be exhibited in many lacunary hyperbolic groups, including Tarski monsters. This represents first examples of groups that have Morse geodesics but no Morse elements. I will describe further properties of non-Morse geodesics and also show how a tree can be quasi-isometrically embedded into such groups.