Cusp transitivity in hyperbolic 3-manifolds

Abstract

In this paper, we study multiply transitive actions of the group of isometries of a cusped finite-volume hyperbolic 3-manifold on the set of its cusps. In particular, we prove a conjecture of Vogeler that there is a largest integer k for which such k-transitive actions exist, and that for each integer $$k \ge 3$$ , there is an upper bound on the possible number of cusps.