NIL-affine crystallographic actions of virtually polycyclic groups

TL;DR

This paper extends the classical affine map correspondence from almost crystallographic groups to virtually polycyclic groups, broadening the understanding of their morphisms and actions on nilpotent Lie groups.

Contribution

It generalizes the affine map induction result to virtually polycyclic groups, introducing methods to analyze translation-like actions and their properties.

Findings

Generalization of affine map induction to virtually polycyclic groups

Development of techniques for studying translation-like actions

Enhanced tools for analyzing morphisms in crystallographic group actions

Abstract

A classical result by K.B. Lee states that every group morphism between almost crystallographic groups is induced by an affine map on the nilpotent Lie group whereon these groups by definition act. It is the main technique for studying morphisms between virtually nilpotent groups, having important applications in fixed point theory, for example as a tool to compute Nielsen numbers for self-maps on infra-nilmanifolds. In this paper we generalize this result to morphisms between virtually polycyclic groups, which are known to act crystallographically by affine transformations on a nilpotent Lie group. The main method is to study the special class of translation-like actions, which behave well under taking subgroups and restricting the action to invariant right cosets.