Cusp types of arithmetic hyperbolic manifolds

TL;DR

This paper characterizes when flat manifolds can serve as cusp cross-sections in arithmetic hyperbolic manifolds, linking the problem to rational representations of holonomy groups and providing criteria based on holonomy and Betti numbers.

Contribution

It establishes necessary and sufficient conditions for flat manifolds to occur as cusp cross-sections in arithmetic hyperbolic manifolds, connecting geometric properties with algebraic representations.

Findings

Flat manifolds with odd order holonomy appear in all commensurability classes if and only if their first Betti number is at least 3.

Examples of flat manifolds that uniquely occur as cusp cross-sections in a single commensurability class.

Pairs of flat manifolds that cannot simultaneously appear as cusp cross-sections in the same quasi-arithmetic hyperbolic manifold.

Abstract

We establish necessary and sufficient conditions for determining when a flat manifold can occur as a cusp cross-section within a given commensurability class of cusped arithmetic hyperbolic manifolds. This reduces the problem of identifying which commensurability classes of arithmetic hyperbolic manifolds can contain a specific flat manifold as a cusp cross-section to a question involving rational representations of the flat manifold's holonomy group. More generally we show that the holonomy representation provides an obstruction on the quasi-arithmetic manifolds containing a given flat manifold as a cusp cross-section. As applications, we prove that a flat manifold $M$ with a holonomy group of odd order appears as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds if and only if $b_1(M)\geq 3$. We also provide examples of flat manifolds that arise as cusp cross-sections in a unique commensurability class of arithmetic hyperbolic manifolds and exhibit examples of pairs of flat manifolds that can never appear as cusp cross-sections in the same quasi-arithmetic hyperbolic manifold.