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Uncountably many maximal-closed subgroups of Sym(N) via reducts of Henson digraphs

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

This work contributes to the two closely related areas of countable homogeneous structures and infinite permutation groups. In the permutation group side, we answered a question of Macpherson that asked to show that there are uncountably many pairwise non-conjugate maximal-closed subgroups of Sym(mathbb{N}). This was achieved by taking the automorphism groups of uncountably many pairwise non-isomorphic Henson digraphs. The fact these groups are maximal-closed follows from the classification of the reducts of Henson digraphs. In itself, this classification contributes to the building list of structures whose reducts are known and also provides further evidence that Thomas’ conjecture is true.

In this talk, my main aim will be to describe the construction of these continuum many maximal-closed subgroups, which will include Henson’s famous construction of continuum many countable homogeneous digraphs. Any remaining time will be spent giving some of the ideas behind how we prove these groups are maximal closed.

This talk is part of the Isaac Newton Institute Seminar Series series.

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