We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the ‘edge’ in our title.) Applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range, a description of the last unstable group up to a single ambiguity, and a lower bound on the rank of the penultimate unstable group. We give similar applications to the general linear groups of the integers and of the field of order 2, this time recovering the known stability range. The results can also be applied to general linear groups of arbitrary principal ideal domains, symmetric groups, and braid groups. Our methods require us to use field coefficients throughout.

中文翻译：

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在稳定范围的边缘

我们证明了配备乘积图的某些群族的一般同调稳定性定理，然后是两个新的定理，提供关于稳定范围外的最后两个同调群的信息。（这最后两个不稳定群是我们标题中的“边缘”。）将我们的结果应用于自由群的自同构群会产生一个新的同源稳定性证明，稳定范围得到改善，对最后一个不稳定群的描述直到一个单一的歧义，以及倒数第二个不稳定组的秩的下界。我们对整数和二阶域的一般线性群给出了类似的应用，这次恢复了已知的稳定性范围。结果也可以应用于任意主理想域、对称群和编织群的一般线性群。