A subgroup determines an equivalence relation on the ambient group, whose classes are cosets. It is sometimes surprising to see that certain theorems about subgroups generalize to arbitrary equivalence relations. I will discuss some examples of this in the 'approximate' setting, including a version of the strong approximation lemma on subgroups of GL_n(F_p) for large p (transposed to connected components of graphs),
Balog-Szemeredi type stabilizer lemmas, and Riemannian homogeneous space models for approximately symmetric approximate equivalence relations.
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