  In this paper, we construct and study derived character maps of
finite-dimensional representations of $\infty$-groups. As models for
$\infty$-groups we take homotopy simplicial groups, i.e. homotopy simplicial
algebras over the algebraic theory of groups (in the sense of Badzioch). We
define cyclic, symmetric and representation homology for `group algebras' over
such groups and construct canonical trace maps relating these homology
theories. In the case of one-dimensional representations, we show that our
trace maps are of topological origin: they are induced by natural maps of
(iterated) loop spaces that are well studied in homotopy theory. Using this
topological interpretation, we deduce some algebraic results about
representation homology: in particular, we prove that the symmetric homology of
group algebras and one-dimensional representation homology are naturally
isomorphic, provided the base ring $k$ is a field of characteristic zero. We
also study the behavior of the derived character maps of $n$-dimensional
representations in the stable limit as $ n\to \infty$, in which case we show
that they `converge' to become isomorphisms.
