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.

中文翻译：

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组中的超交换对和限制它们的3个流形

我们以组为单位介绍了超交换和超交换长度的概念，并表明这些概念与3流形基本群及其Heegaard属的表示密切相关。我们给出了许多示例，包括该行的仿射组中的翻译，并提供了${SL}_2$，与Steinberg关系有关。