Cusp-transitive 4-manifolds with every cusp section

TL;DR

This paper constructs hyperbolic 4-manifolds with transitive cusp symmetries that realize all closed flat 3-manifolds as cusp sections, revealing rich geometric structures and symmetries.

Contribution

It demonstrates that every closed flat 3-manifold can be realized as a cusp section of a hyperbolic 4-manifold with transitive cusp symmetry, including dense metric subsets.

Findings

Realized all closed flat 3-manifolds as cusp sections.

Constructed hyperbolic 4-manifolds with transitive cusp symmetry.

Proved existence of many 4-manifolds with isometric cusps.

Abstract

We realize every closed flat 3-manifold as a cusp section of a complete, finite-volume hyperbolic 4-manifold whose symmetry group acts transitively on the set of cusps. Moreover, for every such 3-manifold, a dense subset of its flat metrics can be realized as cusp sections of a cusp-transitive 4-manifold. Finally, we prove that there are a lot of 4-manifolds with pairwise isometric cusps, for any given cusp type.