New jump operators on equivalence relations

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group $\Gamma$ we introduce the $\Gamma$-jump. We study the elementary properties of the $\Gamma$-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups $\Gamma$, the $\Gamma$-jump is \emph{proper} in the sense that for any Borel equivalence relation $E$ the $\Gamma$-jump of $E$ is strictly higher than $E$ in the Borel reducibility hierarchy. On the other hand there are examples of groups $\Gamma$ for which the $\Gamma$-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 infinite $\Gamma$-tree, and relate these to iterates of the $\Gamma$-jump. We also produce several new examples of equivalence relations that arise from applying the $\Gamma$-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.