Continuous isomorphisms between groups definable in o-minimal expansions of the real field

TL;DR

This paper investigates when Lie isomorphisms between definable groups in o-minimal structures are themselves definable, extending known results and providing conditions for preserving o-minimality and analytic structures.

Contribution

It characterizes when Lie isomorphisms between definable groups are definable and generalizes Wilkie's o-minimality result for the exponential function.

Findings

Characterization of when Lie isomorphisms are definable in o-minimal structures

Extension of Wilkie's result on o-minimality of the exponential function

Definable groups can be given an analytic manifold structure

Abstract

In this paper we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field, which we will refer to as ``definable groups''. With this terminology, it is known (\cite{Pi88}) that any definable group is a Lie group, and in \cite{COP} a complete characterization of when a Lie group is \emph{Lie isomorphic} to a definable group'' was given. We continue the analysis by explaining when a Lie isomorphism between definable groups is definable. Among other things, we generalize Wilkie's result on the o-minimality of the exponential function (\cite{Wilkie}) by completely characterizing when, given an o-minimal expansion $\mathcal R$ of the real field and a Lie isomorphisms $\phi$ between two $\mathcal R$-definable groups $G_1, G_2$, $\phi$ can be added to the language of $\mathcal R$ preserving o-minimality. We also prove that any definable group $G$ can be endowed with an analytic manifold structure definable in $\mathcal R_{\text{Pfaff}}$ that makes it an analytic group.