Compactifications of pseudofinite and pseudo-amenable groups
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| Format: | Preprint |
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2023
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| _version_ | 1866909650473975808 |
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| author | Conant, Gabriel Hrushovski, Ehud Pillay, Anand |
| author_facet | Conant, Gabriel Hrushovski, Ehud Pillay, Anand |
| contents | We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing's work, the Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan \cite{Kazh} on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudo-amenable groups to compact Lie groups. Together with the stabilizer theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative) analogue of Bogolyubov's Lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_08440 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Compactifications of pseudofinite and pseudo-amenable groups Conant, Gabriel Hrushovski, Ehud Pillay, Anand Logic Group Theory We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing's work, the Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan \cite{Kazh} on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudo-amenable groups to compact Lie groups. Together with the stabilizer theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative) analogue of Bogolyubov's Lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups. |
| title | Compactifications of pseudofinite and pseudo-amenable groups |
| topic | Logic Group Theory |
| url | https://arxiv.org/abs/2308.08440 |