Continuum Percolation at and above the Uniqueness Threshold on Homogeneous Spaces

We consider the Poisson Boolean model of continuum percolation on a homogeneous space M. Let λ be the intensity of the underlying Poisson process. Let λ _u be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if λ>λ _u , then there is a.s. a unique unbounded component at λ. Then we let M=?²×? and show that at λ _u there is a.s. not a unique unbounded component. These results are continuum analogs of theorems by Häggström, Peres and Schonmann.