Cusp transitivity in hyperbolic 3-manifolds

TL;DR

This paper investigates the symmetry actions on cusps of hyperbolic 3-manifolds, proving a conjecture about the maximum transitivity and bounds on cusps for certain symmetries.

Contribution

It proves a conjecture regarding the maximum transitivity of cusp actions and establishes bounds on the number of cusps for higher transitivity levels.

Findings

Existence of a maximum transitivity level for cusp actions

Upper bounds on the number of cusps for k-transitive actions

Confirmation of Vogeler's conjecture

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 $k$ for which such $k$-transitive actions exist, and that for each $k \geq 3$, there is an upper bound on the possible number of cusps.