Title
The connected component of groups in n-dependent theories
Abstract
Inspired by its analog in algebraic geometry, we define the connected component over a parameter set A of a group to be the intersection of all A-definable subgroups of finite index. Moreover, this notion has corresponding versions for type-definable and invariant subgroups. In general, these groups get smaller and smaller while the parameter set grows. If this is not the case, i. e. the connected component over A corresponds to the one over the empty set, we say that the connect component exists.
By results of Shelah and Gismatulin we know that all three connected components exists in NIP groups. NIP theories, also called dependent or 1-dependent theories, is the first class of the hierarchy of n-dependent theories. This hierarchy is strict and every n-dependent theory is also n+1 dependent. Shelah obtained a weaker result for the type-definable connected component for 2-dependent groups.
We will first discuss the existence of strictly n-dependent groups, gives some explicit examples and finally present a version of Shelah’s result for n-dependent theories.’
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