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

Alf Onshuus

TL;DR

This work investigates when Lie isomorphisms between groups definable in o-minimal expansions of the real field are definable. The authors identify the maximal torsion-free normal definable subgroup $U_G$ and near-central subgroups as the main obstruction to definability, and show that a surjective Lie homomorphism $\\alpha$ is definable provided its restriction to near-central torsion-free subgroups of the radical is definable; if $\\alpha$ maps $U_1$ to $U_2$, definability occurs in the Pfaffian closure $\\mathcal{R}_{\\text{Pfaff}}$. The results yield that any definable group can be endowed with an analytic manifold structure definable in $\\mathcal{R}_{\\text{Pfaff}}$, connecting to COP’s characterization of Lie-isomorphisms to definable groups and Wilkie-type o-minimality enhancements. The paper also develops a framework for adding Lie isomorphisms to the language while preserving o-minimality and discusses sharp-o-minimality and potential effective bounds, including generalized exponentials and Pfaffian-solution closures. Overall, the results extend the o-minimality of exponentials to a broader class of Lie isomorphisms and provide structural tools for translating between Lie-theoretic and definable-group worlds.

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 $φ$ between two $\mathcal R$-definable groups $G_1, G_2$, $φ$ 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.

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

TL;DR

This work investigates when Lie isomorphisms between groups definable in o-minimal expansions of the real field are definable. The authors identify the maximal torsion-free normal definable subgroup and near-central subgroups as the main obstruction to definability, and show that a surjective Lie homomorphism is definable provided its restriction to near-central torsion-free subgroups of the radical is definable; if maps to , definability occurs in the Pfaffian closure . The results yield that any definable group can be endowed with an analytic manifold structure definable in , connecting to COP’s characterization of Lie-isomorphisms to definable groups and Wilkie-type o-minimality enhancements. The paper also develops a framework for adding Lie isomorphisms to the language while preserving o-minimality and discusses sharp-o-minimality and potential effective bounds, including generalized exponentials and Pfaffian-solution closures. Overall, the results extend the o-minimality of exponentials to a broader class of Lie isomorphisms and provide structural tools for translating between Lie-theoretic and definable-group worlds.

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 of the real field and a Lie isomorphisms between two -definable groups , can be added to the language of preserving o-minimality. We also prove that any definable group can be endowed with an analytic manifold structure definable in that makes it an analytic group.
Paper Structure (11 sections, 11 theorems, 11 equations)

This paper contains 11 sections, 11 theorems, 11 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Preliminaries
  3. Proof of Theorem \ref{['Theorem']}
  4. The restriction of $\alpha$ to $R$ is definable.
  5. Definability of the restriction of $\alpha$ to $L\cdot Z^0$.
  6. Proof of Theorem \ref{['Theorem']}
  7. Adding the Lie isomorphisms between definable groups in expansions of the real field.
  8. Lie isomorphisms between definable torsion free abelian subgroups
  9. Compact abelian subgroups
  10. Proof of Theorem \ref{['ThmHomomorphism']}
  11. Further considerations. sharp-o-minimality

Key Result

Theorem 1.1

Assume that $\mathcal{R}$ includes the exponential function. Let $\alpha$ be a surjective Lie homomorphism between two definable groups $G_1$ and $G_2$, and assume that the restriction of $\alpha$ to any definable abelian subgroup of $R_{G_1}$ is definable. Then $\alpha$ is definable.

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.6
  • Definition 1.11
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Claim 2.4
  • ...and 22 more