  The Kac model describes the local evolution of a gas of $N$ particles with
three dimensional velocities by a random walk in which the steps correspond to
binary collisions that conserve momentum as well as energy. The state space of
this walk is a sphere of dimension $3N - 4$. The Kac conjecture concerns the
spectral gap in the one step transition operator $Q$ for this walk. In this
paper, we compute the exact spectral gap.
  As in previous work by Carlen, Carvalho and Loss where a lower bound on the
spectral gap was proved, we use a method that relates the spectral properties
of $Q$ to the spectral properties of a simpler operator $P$, which is simply an
average of certain non commuting projections. The new feature is that we show
how to use a knowledge of certain eigenfunctions and eigenvalues of $P$ to
determine spectral properties of $Q$, instead of simply using the spectral gap
for $P$ to bound the spectral gap for $Q$, inductively in $N$, as in previous
work. The methods developed here can be applied to many other high--dimensional
stochastic process, as we shall explain.
  We also use some deep results on Jacobi polynomials to obtain the required
spectral information on $P$, and we show how the identity through which Jacobi
polynomials enter our problem may be used to obtain new bounds on Jacobi
polynomials.
