Vertical perimeter versus horizontal perimeter

The discrete Heisenberg group $\mathbb{H}{\mathbb{Z}}^{2k+1}$ is the group generated by $a1,b1,\ldots,ak,bk,c$, subject to the relations $[a1,b1]=\ldots=[ak,bk]=c$ and $[ai,aj]=[bi,bj]=[ai,bj]=[ai,c]=[bi,c]=1$ for every distinct $i,j\in {1,\ldots,k}$. Denote $S={a1^{\pm 1},b1^{\pm 1},\ldots,ak^{\pm 1},bk^{\pm 1}}$. The horizontal boundary of $\Omega\subset \mathbb{H}{\mathbb{Z}}^{2k+1}$, denoted $\partial{h}\Omega$, is the set of all $(x,y)\in \Omega\times (\mathbb{H}{\mathbb{Z}}^{{2k+1}\setminus} \Omega)$ such that $x^{-1}y\in S$. The horizontal perimeter of $\Omega$ is $|\partial{h}\Omega|$. For $t\in \mathbb{N}$, define $\partial^t{v} \Omega$ to be the set of all $(x,y)\in \Omega\times (\mathbb{H}{\mathsf{Z}}^{{2k+1}\setminus} \Omega)$ such that $x^{-1}y\in {c^t,c{-t}}$. The vertical perimeter of $\Omega$ is defined by $|\partial{v}\Omega|= \sqrt{\sum{t=1}^\infty |\partial^t{v}\Omega|^2/t2}$. It is shown here that if $k\ge 2$, then $|\partial{v}\Omega|\lesssim \frac{1}{k} |\partial{h}\Omega|$. The proof of this "vertical versus horizontal isoperimetric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an "intrinsic corona decomposition." This allows one to deduce an endpoint $W^{1,1}\to L2(L1)$ boundedness of a certain singular integral operator from a corresponding lower-dimensional $W^{1,2}\to L2(L2)$ boundedness. The above inequality has several applications, including that any embedding into $L1$ of a ball of radius $n$ in the word metric on $\mathbb{H}{\mathbb{Z}}^{5}$ incurs bi-Lipschitz distortion that is at least a constant multiple of $\sqrt{\log n}$. It follows that the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut Problem on inputs of size $n$ is at least a constant multiple of $\sqrt{\log n}$.