  Mulmuley and Sohoni (GCT1 in SICOMP 2001, GCT2 in SICOMP 2008) proposed to
view the permanent versus determinant problem as a specific orbit closure
problem and to attack it by methods from geometric invariant and representation
theory. We adopt these ideas towards the goal of showing lower bounds on the
border rank of specific tensors, in particular for matrix multiplication. We
thus study specific orbit closure problems for the group $G = GL(W_1)\times
GL(W_2)\times GL(W_3)$ acting on the tensor product $W=W_1\otimes W_2\otimes
W_3$ of complex finite dimensional vector spaces. Let $G_s = SL(W_1)\times
SL(W_2)\times SL(W_3)$. A key idea from GCT2 is that the irreducible
$G_s$-representations occurring in the coordinate ring of the $G$-orbit closure
of a stable tensor $w\in W$ are exactly those having a nonzero invariant with
respect to the stabilizer group of $w$.
  However, we prove that by considering $G_s$-representations, as suggested in
GCT1-2, only trivial lower bounds on border rank can be shown. It is thus
necessary to study $G$-representations, which leads to geometric extension
problems that are beyond the scope of the subgroup restriction problems
emphasized in GCT1-2. We prove a very modest lower bound on the border rank of
matrix multiplication tensors using $G$-representations. This shows at least
that the barrier for $G_s$-representations can be overcome. To advance, we
suggest the coarser approach to replace the semigroup of representations of a
tensor by its moment polytope. We prove first results towards determining the
moment polytopes of matrix multiplication and unit tensors.
