Title: Morse geodesics in torsion groups,
Abstract: In this paper we exhibit Morse geodesics, often called "hyperbolic
directions", in infinite unbounded torsion groups. The groups studied are
lacunary hyperbolic groups and constructed using graded small cancellation
conditions. In all previously known examples, Morse geodesics were found in
groups which also contained Morse elements, infinite order elements whose
cyclic subgroup gives a Morse quasi-geodesic. Our result presents the first
example of a group which contains Morse geodesics but no Morse elements. In
fact, we show that there is an isometrically embedded $7$-regular tree inside
such groups where every infinite, simple path is a Morse geodesic.