A matrix theory for finite group actions

We introduce a new matrix theory to investigate finite group actions on spa ces. Given a finite group action, we associate it with a family of orbit ma trices. The spectral radius of an action is also introduced. It is shown th at the spectral raduis is bounded below by a constant depending only on som e geometric invariants of the underlying Riemannian manifolds. The relation between the eigenspaces of orbit matrices and regular representations of f inite groups are also investigated. In particular, we obtain that the eigen values of orbit matrices reveal some structures of the groups.