Topological groups, μ-types and their stabilizers

2014/09/18 by Ya’acov Peterzil, Peterzil, Ya'Acov, Sergei Starchenko +1
Mathematics · #03C64 #03C98 #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras

paper · doi:10.48550/arxiv.1409.5355

openalex publication_date 2014/09/18 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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^μ_G(M)=\p_μ:p∈ 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_μ(p), which is the stabilizer of p_μ. 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^μ_G(M) and its connection to the Samuel compactification of topological groups.

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