Vertical perimeter versus horizontal perimeter

TL;DR

This paper establishes a new isoperimetric inequality in the discrete Heisenberg group relating vertical and horizontal perimeters, with implications for metric embeddings and approximation algorithms.

Contribution

It introduces a novel structural decomposition of sets in the Heisenberg group and proves a vertical versus horizontal perimeter inequality for higher dimensions.

Findings

Vertical perimeter is bounded by horizontal perimeter divided by dimension parameter k.

Embedding distortion of Heisenberg group balls into L1 grows at least as sqrt(log n).

Lower bounds on semidefinite programming gaps for Sparsest Cut problem.

Abstract

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 $\sqrt{\log n}$.