Invariantly universal analytic quasi-orders

We introduce the notion of an invariantly universal pair (S;E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sucient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in dierent areas of mathematics, when they are paired with natural equivalence relations.