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In this paper, we establish a precise connection between higher rhoinvariants and delocalized eta invariants. Given an element in a discretegroup, if its conjugacy class has polynomial growth, then there is a naturaltrace map on the $K_0$-group of its group $C^\ast$-algebra. For each such tracemap, we construct a determinant map on secondary higher invariants. We showthat, under the evaluation of this determinant map, the image of a higher rhoinvariant is precisely the corresponding delocalized eta invariant of Lott. Asa consequence, we show that if the Baum-Connes conjecture holds for a group,then Lott's delocalized eta invariants take values in algebraic numbers. Wealso generalize Lott's delocalized eta invariant to the case where thecorresponding conjugacy class does not have polynomial growth, provided thatthe strong Novikov conjecture holds for the group.

Delocalized eta invariants, algebraicity, and $K$-theory of group $C^*$-algebras