  The discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{2k+1}$ is the group
generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations
$[a_1,b_1]=\ldots=[a_k,b_k]=c$ and
$[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every distinct $i,j\in
\{1,\ldots,k\}$. Denote $S=\{a_1^{\pm 1},b_1^{\pm 1},\ldots,a_k^{\pm
1},b_k^{\pm 1}\}$. 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
$|\partial_{v}\Omega|= \sqrt{\sum_{t=1}^\infty |\partial^t_{v}\Omega|^2/t^2}$.
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 $W^{1,1}\to L_2(L_1)$ boundedness of a certain singular integral
operator from a corresponding lower-dimensional $W^{1,2}\to L_2(L_2)$
boundedness. The above inequality has several applications, including that any
embedding into $L_1$ 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 $\sqrt{\log n}$.
