Sergio Fenley
Florida State University
March 27, 2025
Existence of quasigeodesic Anosov flows in hyperbolic 3-manifolds: A quasigeodesic in a manifold is a curve so that when lifted to the universal cover is uniformly efficient up to a bounded multiplicative and added error in measuring length. A flow is quasigeodesic if all flow lines are quasigeodesics. We prove that an Anosov flow in a closed hyperbolic manifold is quasigeodesic if and only if it is not R-covered. Here R-covered means that the stable 2-dim foliation of the flow, lifts to a foliation in the universal cover whose leaf space is homeomorphic to the real numbers. There are many examples of quasigeodesic Anosov flows in closed hyperbolic 3-manifolds. There are consequences for the continuous extension property of Anosov foliations, and the existence of group invariant Peano curves associated with Anosov flows.
Florida State University
March 27, 2025
Existence of quasigeodesic Anosov flows in hyperbolic 3-manifolds: A quasigeodesic in a manifold is a curve so that when lifted to the universal cover is uniformly efficient up to a bounded multiplicative and added error in measuring length. A flow is quasigeodesic if all flow lines are quasigeodesics. We prove that an Anosov flow in a closed hyperbolic manifold is quasigeodesic if and only if it is not R-covered. Here R-covered means that the stable 2-dim foliation of the flow, lifts to a foliation in the universal cover whose leaf space is homeomorphic to the real numbers. There are many examples of quasigeodesic Anosov flows in closed hyperbolic 3-manifolds. There are consequences for the continuous extension property of Anosov foliations, and the existence of group invariant Peano curves associated with Anosov flows.