Ann. of Math. (2) 188 (1), 171-279, (July 2018) DOI: 10.4007/annals.2018.188.1.4
KEYWORDS: Heisenberg group, Isoperimetric inequalities, metric embeddings, Sparsest Cut Problem, approximation algorithms, semidefinite programming, metrics of negative type, 46B85, 20F65, 30L055, 68W25, 43A15

Given $k\in \mathbb{N}$, the $k$'th discrete Heisenberg group, denoted $ \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}$, is the group generated by the elements $a_1,b_1,\ldots,a_k,b_k,c$, subject to the commutator relations $[a_1,b_1]=\cdots=[a_k,b_k]=c$, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct $i,j\in \{1,\dots,k\}$, we have $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$. (In particular, this implies that $c$ is in the center of $\mathbb{H}_{ \scriptscriptstyle{\mathbb{Z}}}^{2k+1}$.) Denote $\mathfrak{S}_k=\{a_1,b_1,a_1^{-1},b_1^{-1},\ldots,a_k,b_k,a_k^{-1},b_k^{-1}\}$. The * horizontal boundary* of $\Omega\subseteq \mathbb{H}_{ \scriptscriptstyle{ \mathbb{Z}}}^{2k+1}$, denoted $\partial_{\mathsf{h}}\Omega$, is the set of all those pairs $(x,y)\in \Omega\times (\mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in \mathfrak{S}_k$. The * horizontal perimeter* of $\Omega$ is the cardinality $|\partial_{\mathsf{h}}\Omega|$ of $\partial_{\mathsf{h}}\Omega$; i.e., it is the total number of edges incident to $\Omega$ in the Cayley graph induced by $\mathfrak{S}_k$. For $t\in \mathbb{N}$, define $\partial^t_{\mathsf{v}} \Omega$ to be the set of all those pairs $(x,y)\in \Omega\times (\mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in \{c^t,c^{-t}\}$. Thus, $|\partial^t_{\mathsf{v}}\Omega|$ is the total number of edges incident to $\Omega$ in the (disconnected) Cayley graph induced by $\{c^t,c^{-t}\}\subseteq \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}$. The * vertical perimeter * of $\Omega$ is defined by $|\partial_{\mathsf{v}}\Omega|= \sqrt{\sum_{t=1}^\infty |\partial^t_{\mathsf{v}}\Omega|^2/t^2}$. It is shown here that if $k\ge 2$, then $|\partial_{\mathsf{v}}\Omega|\lesssim \frac{1}{k} |\partial_{\mathsf{h}}\Omega|$. 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. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every $n\in \mathbb{N}$, any embedding into an $L_1(\mu)$ space of a ball of radius $n$ in the word metric on $ \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{5}$ that is induced by the generating set $\mathfrak{S}_2$ incurs bi-Lipschitz distortion that is at least a universal constant multiple of $\sqrt{\log n}$. As an application to approximation algorithms, it follows that for every $n\in \mathbb{N}$, the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut Problem on inputs of size $n$ is at least a universal constant multiple of $\sqrt{\log n}$.M