GL(4,R) representation of the gravity action on the piecewise flat spacetime

The gravity action on the piecewise flat Riemannian manifold is formulated using the discrete set of the nondegenerate 4$\times$4 matrices on the 3-simplices as some connection type variables. These variables are the discrete counterpart of the affine (Christoffel) connection used as independent variables in the Palatini form of the Einstein gravity action. Excluding these with the help of the equations of motion we get the original discrete gravity action on the piecewise flat spacetime (Regge action). The discrete version of the diffeomorphisms and path integral are briefly discussed.