Abstract
Abstract A D-quasirandom group is a group without any non-trivial unitary representation of dimension less than D. Given a sequence of groups with increasing quasirandomness, it is natural to ask if the ultraproduct will end up with no finite dimensional unitary representation at all. This is not true in general, but we answer this question in the affirmative when the groups in question have uniform small cosocles, i.e., their quotients by small kernels are direct products of finite simple groups. Two applications of our results are given, one in triangle patterns inside quasirandom groups and one in self-bohrifying groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups.
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