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We introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

We prove that for a finitely generated infinite nilpotent group G with structure (G, ·, …), the connected component G*0 of a sufficiently saturated extension G* of G exists and equals

We construct an expansion of ℤ by a predicate (ℤ, +, P) such that the type-connected component is strictly smaller than ℤ*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem for finite partitions of groups.

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ON MODEL-THEORETIC CONNECTED GROUPS

Model theoretic connected components of finitely generated nilpotent groups

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2 results

ON MODEL-THEORETIC CONNECTED GROUPS

Model theoretic connected components of finitely generated nilpotent groups