Rectifiable and flat G chains in a metric space

Chains, in a complete metric space, which have coefficients in a normed abelian group $G$ are studied. An $m$ dimensional rectifiable chain is the Lipschitz push-forward of a region in ${\Bbb R}^m$ equipped with a measurable $G$-valued density. Flat chains are obtained by completion using a certain flat norm on polyhedral or Lipschitz chain approximations. Numerous basic results of geometric measure theory for these chains are derived including the rectifiability of finite mass flat chains provided that $G$ contains no nonconstant Lipschitz curves. The work here generalizes, and uses many ideas from, the 1999 paper of B. White on at $G$ chains in ${\Bbb R}^n$ and the 2000 paper of L. Ambrosio and B. Kirchheim on currents in a metric space. The use of flat and rectifiable chains in geometric variational problems or in defining geometric homology theories may reveal geometric properties of spaces.