We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group $\mathrm{\Gamma }$ we introduce the $\mathrm{\Gamma }$-jump. We study the elementary properties of the $\mathrm{\Gamma }$-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups $\mathrm{\Gamma }$, the $\mathrm{\Gamma }$-jump is proper in the sense that for any Borel equivalence relation $E$ the $\mathrm{\Gamma }$-jump of $E$ is strictly higher than $E$ in the Borel reducibility hierarchy. On the other hand, there are examples of groups $\mathrm{\Gamma }$ for which the $\mathrm{\Gamma }$-jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the full $\mathrm{\Gamma }$-tree, and relate these to iterates of the $\mathrm{\Gamma }$-jump. We also produce several new examples of equivalence relations that arise from applying the $\mathrm{\Gamma }$-jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank.

我们在 Borel 等价关系上引入了一个新的跳跃算子族；具体来说，对于每个可数组$\mathrm{\Gamma }$我们介绍$\mathrm{\Gamma }$-跳。我们研究的基本性质$\mathrm{\Gamma }$-jumps 并将它们与之前研究过的其他跳跃运算符进行比较。我们的主要成果之一是为许多群体建立$\mathrm{\Gamma }$， 这$\mathrm{\Gamma }$-jump 是适当的，因为对于任何 Borel 等价关系$乙$这$\mathrm{\Gamma }$-跳跃$乙$严格高于$乙$在 Borel 还原性层次结构中。另一方面，有组的例子$\mathrm{\Gamma }$为此$\mathrm{\Gamma }$-jump 不合适。为了建立适当性，我们分析了由我们表示为完整的自同构群的连续作用引起的 Borel 等价关系$\mathrm{\Gamma }$-tree，并将这些与$\mathrm{\Gamma }$-跳。我们还产生了几个等价关系的新例子，这些例子源于应用$\mathrm{\Gamma }$-跳转到经典研究的等价关系并推导出与这些相关的一般遍历性结果。我们应用我们的结果表明，可数散乱线性阶的同构问题的复杂性随着等级的增加而适当增加。