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We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3‐manifolds and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in    SL  2   , related to the Steinberg relation.

Overcommuting pairs in groups and 3‐manifolds bounding them