Cusp cross-sections of low-dimensional arithmetic hyperbolic manifolds

Abstract

We study which compact flat manifolds of dimensions $3 \leq n \leq 6$ occur as cusp cross-sections in commensurability classes of non-compact arithmetic hyperbolic $(n+1)$-manifolds. Using the classification of low-dimensional flat manifolds together with the relationship between their holonomy representations and rational quadratic forms, we determine the possible arithmetic commensurability classes for every flat manifold in these dimensions. We show that every flat $n$-manifold for $n=3,4,5$ occurs as a cusp cross-section in infinitely many distinct arithmetic commensurability classes. In dimension six, the same holds with exactly eight exceptions: eight non-orientable flat $6$-manifolds occur in a unique arithmetic commensurability class. We also show that, for every $3 \leq n \leq 6$, there is a single commensurability class of arithmetic hyperbolic $(n+1)$-manifolds containing every flat $n$-manifold as a cusp cross-section.