Topological groups, $\mu$-types and their stabilizers

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, $S_G^{\mu }(M)=\{p_{\mu }:p\in S_G(M)\}$ 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 $S_G^{\mu }(M)$ and its connection to the Samuel compactification of topological groups.