O-minimal flows on nilmanifolds

TL;DR

This paper characterizes the topological closures of images of semi-algebraic and definable sets under the quotient map in nilmanifolds, revealing their structure via algebraic subgroups and establishing equidistribution for curves.

Contribution

It provides a detailed description of closures of definable sets in nilmanifolds and introduces an equidistribution theorem for curves within this framework.

Findings

Closure of definable sets described by finitely many algebraic cosets.

Closure description is independent of the lattice $\Gamma$.

Established equidistribution for curves in nilmanifolds.

Abstract

Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $\pi(X)$ in the compact nilmanifold $G/\Gamma$. Our theorem describes $\mathrm{cl}(\pi(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $\Gamma$. We also prove an equidistribution result in the case of curves.