 Amenable boundary growth of lattice boxes

For the nearest-neighbour graph on ${\mathbb{Z}}^d$, a box $B_n=[-n,n{]}^d\cap {\mathbb{Z}}^d$ has $(2n+1{)}^d$ graph vertices and outer graph vertex boundary of size $2d(2n+1{)}^{d-1}$. Thus $|{\partial }^{+}{B}_n|/|B_n|\to 0$. This boundary-to-volume property is the geometric mechanism that contradicts a positive volume density of trifurcation points in the uniqueness proof. It realizes the boundary-growth property of the translation amenable group on the lattice.