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Cut-and-paste $K$-theory has recently emerged as an important variant ofhigher algebraic $K$-theory. However, many of the powerful tools used to studyclassical higher algebraic $K$-theory do not yet have analogues in thecut-and-paste setting. In particular, there does not yet exist a sensiblenotion 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 grouphomology, and use this trace to prove the existence of some nonzero classes inthe 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 withhomotopy colimits, but we give a self-contained proof. We then use this resultto re-interpret the trace map as a partial inverse to the map that commuteshomotopy orbits with algebraic $K$-theory.

A trace map on higher scissors congruence groups