Lie groups as permutation groups: Ulam's problem in the nilpotent case

TL;DR

This paper proves that connected nilpotent Lie groups can be represented on countable structures, advancing understanding of Ulam's problem beyond the linear case and highlighting the importance of nilpotent groups.

Contribution

It establishes the first non-linear case solution for Ulam's problem, specifically for connected nilpotent Lie groups.

Findings

Connected nilpotent Lie groups can be represented on countable structures

This extends known results from the linear case to a broader class of groups

Highlights the significance of nilpotent groups in Ulam's problem

Abstract

Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discussed to illustrate the relevance of nilpotent groups for Ulam's problem.