Vertical perimeter versus horizontal perimeter

Given k ∈ N, the k’th discrete Heisenberg group, denoted H Z , is the group generated by the elements a1, b1, . . . , ak, bk, c, subject to the commutator relations [a1, b1] = . . . = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ {1, . . . , k} we have [ai, aj ] = [bi, bj ] = [ai, bj ] = [ai, c] = [bi, c] = 1 (in particular, this implies that c is in the center of H Z ). Denote Sk = {a1, b1, a−1 1 , b −1 1 , . . . , ak, bk, a −1 k , b −1 k }. The horizontal boundary of Ω ⊆ H Z , denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω×(H Z rΩ) such that x−1y ∈ Sk. The horizontal perimeter of Ω is the cardinality |∂hΩ| of ∂hΩ, i.e., it is the total number of edges incident to Ω in the Cayley graph induced by Sk. For t ∈ N, define ∂ vΩ to be the set of all those pairs (x, y) ∈ Ω × (H Z r Ω) such that x−1y ∈ {c, c−t}. Thus, |∂ vΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by {c, c−t} ⊆ H Z . The vertical perimeter of Ω is defined by |∂vΩ| = √∑∞ t=1 |∂ vΩ|/t. It is shown here that if k > 2, then |∂vΩ| . 1 k |∂hΩ|. 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 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W 1,2 → L2(L2) boundedness. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every n ∈ N, any embedding into an L1(μ) space of a ball of radius n in the word metric on HZ that is induced by the generating set S2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ logn. As an application to approximation algorithms, it follows that for every n ∈ 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 √ logn.