Topological groups, \mu-types and their stabilizers

TL;DR

This paper introduces a new topological space of partial types for definable topological groups, linking model theory with topological group theory and analyzing stabilizers and compactifications.

Contribution

It constructs a compact $G$-space of partial types associated with definable topological groups and explores the properties of stabilizers and their relation to Samuel compactification.

Findings

The space $S^mu_G(M)$ is a compact $G$-space derived from types.

Definable types have stabilizers that are torsion-free solvable groups.

The construction connects model-theoretic types with topological group compactifications.

Abstract

We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To each such group $G$ we associate a compact $G$-space of partial types $S^\mu_G(M)=\{p_\mu:p\in S_G(M)\}$ which is the quotient of the usual type space $S_G(M)$ by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if $p$ is a definable type then it has a corresponding definable subgroup $Stab_\mu(p)$, which is the stabilizer of $p_\mu$. This group is nontrivial when $p$ is unbounded in the sense of $\mathcal M$; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of $S^\mu_G(M)$ and its connection to the Samuel compactification of topological groups.