Cusp transitivity in hyperbolic 3-manifolds

TL;DR

This paper investigates the symmetry actions on cusps of hyperbolic 3-manifolds, demonstrating the existence of manifolds with various levels of cusp transitivity and constructing examples with unbounded cusps for certain symmetry levels.

Contribution

It constructs explicit examples of hyperbolic 3-manifolds with specified cusp transitivity levels and shows that for $k=2$, there is no upper bound on the number of cusps.

Findings

Existence of manifolds with $k=1,2,4$ cusp transitivity

Construction of manifolds with unbounded cusps for $k=2$

Analysis of symmetry actions on hyperbolic 3-manifolds

Abstract

Let $M$ be a cusped finite-volume hyperbolic three-manifold with isometry group $G$. Then $G$ induces a $k$-transitive action by permutation on the cusps of $M$ for some integer $k\ge 0$. Generically $G$ is trivial and $k=0$, but $k>0$ does occur in special cases. We show examples with $k=1,2,4$. An interesting question concerns the possible number of cusps for a fixed $k$. Our main result provides an answer for $k=2$ by constructing a family of manifolds having no upper bound on the number of cusps.