Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order

TL;DR

This paper investigates how uniquely the fundamental groups of certain flat manifolds are determined by their finite quotients, providing a formula for their profinite genus when the holonomy group is cyclic of square-free order.

Contribution

It introduces a formula for the profinite genus of fundamental groups of flat manifolds with cyclic, square-free holonomy groups, advancing understanding of their algebraic structure.

Findings

Derived a formula for the profinite genus

Characterized the extent of distinguishability by finite quotients

Focused on manifolds with cyclic, square-free holonomy groups

Abstract

In this article we study the extent to which an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order.