  Cut-and-paste $K$-theory has recently emerged as an important variant of
higher algebraic $K$-theory. However, many of the powerful tools used to study
classical higher algebraic $K$-theory do not yet have analogues in the
cut-and-paste setting. In particular, there does not yet exist a sensible
notion of the Dennis trace for cut-and-paste $K$-theory.
  In this paper we address the particular case of the $K$-theory of polyhedra,
also called scissors congruence $K$-theory. We introduce an explicit,
computable trace map from the higher scissors congruence groups to group
homology, and use this trace to prove the existence of some nonzero classes in
the higher scissors congruence groups.
  We also show that the $K$-theory of polyhedra is a homotopy orbit spectrum.
This fits into Thomason's general framework of $K$-theory commuting with
homotopy colimits, but we give a self-contained proof. We then use this result
to re-interpret the trace map as a partial inverse to the map that commutes
homotopy orbits with algebraic $K$-theory.
