  We prove that scissors congruence $K$-theory is a Thom spectrum, whose base
is the homotopy orbit space of a flag complex. We use this to show that the
higher $K$-groups are rationally, and in the Euclidean case integrally, the
homology of the isometry group with Steinberg module coefficients. This allows
us to make the first calculations of these groups above $K_1$.
