Measure equivalence and coarse equivalence for unimodular locally compact groups

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This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they admit free, ergodic, probability measure preserving actions whose cross section equivalence relations are stably orbit equivalent. Using this we prove that in the presence of amenability any two such groups are measure equivalent and that both amenability and property (T) are preserved under measure equivalence, extending results of Connes–Feldman–Weiss and Furman. Furthermore, we introduce a notion of uniform measure equivalence for unimodular, locally compact, second countable groups, and prove that under the additional assumption of amenability this notion coincides with coarse equivalence, generalizing results of Shalom and Sauer. Throughout the article we rigorously treat measure theoretic issues arising in the setting of non-discrete groups.