  In this paper, we study lower bounds on the K-theory of the maximal
$C^*$-algebra of a discrete group based on the amount of torsion it contains.
We call this the finite part of the operator K-theory and give a lower bound
that is valid for a large class of groups, called the "finitely embeddable
groups". The class of finitely embeddable groups includes all residually finite
groups, amenable groups, Gromov's monster groups, virtually torsion free groups
(e.g. $Out(F_n)$), and any group of analytic diffeomorphisms of an analytic
connected manifold fixing a given point. It is an open question if every
countable group is finitely embeddable. We apply this result to measure the
degree of non-rigidity for any compact oriented manifold $M$ with dimension
$4k-1$ $(k>1)$. We derive a lower bound on the rank of the structure group
$S(M)$ in this case. For a compact Riemannian manifold $M$ with dimension
greater than or equal to 5 and positive scalar curvature metric, there is an
abelian group $P(M)$ that measures the size of the space of all positive scalar
curvature metrics on $M$. We obtain a lower bound on the rank of the abelian
group $P(M)$ when the compact smooth spin manifold $M$ has dimension $2k-1$
$(k>2)$ and the fundamental group of $M$ is finitely embeddable.
