# Vertical perimeter versus horizontal perimeter

Assaf Naor, Robert Young

Research output: Contribution to journalArticle

### Abstract

Given k ∈ N, the k'th discrete Heisenberg group, denoted H 2k+1, 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 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ H 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(H 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. 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 b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (H 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ H 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. "This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → 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 H 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity 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 √ log n.

Original language English (US) 171-279 109 Annals of Mathematics 188 1 https://doi.org/10.4007/annals.2018.188.1.4 Published - Jul 1 2018

### Fingerprint

Perimeter
Isoperimetric
Horizontal
Generating Set
Vertical
Cayley Graph
Heisenberg Group
Boundedness
Decompose
Semidefinite Program
Corona
Singular Integral Operator
Discrete Group
Commute
Commutator
Lipschitz
Approximation Algorithms
Deduce
Cardinality
Ball

### Keywords

• Approximation algorithms
• Heisenberg group
• Isoperimetric inequalities
• Metric embeddings
• Metrics of negative type
• Semidefinite programming
• Sparsest Cut Problem

### ASJC Scopus subject areas

• Statistics and Probability
• Statistics, Probability and Uncertainty

### Cite this

In: Annals of Mathematics, Vol. 188, No. 1, 01.07.2018, p. 171-279.

Research output: Contribution to journalArticle

Naor, Assaf ; Young, Robert. / Vertical perimeter versus horizontal perimeter. In: Annals of Mathematics. 2018 ; Vol. 188, No. 1. pp. 171-279.
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abstract = "Given k ∈ N, the k'th discrete Heisenberg group, denoted Hℤ 2k+1, 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 Hℤ 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ Hℤ 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(Hℤ 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. 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 b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (Hℤ 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ Hℤ 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality{"} uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. {"}This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → 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 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity 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 √ log n.",
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N2 - Given k ∈ N, the k'th discrete Heisenberg group, denoted Hℤ 2k+1, 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 Hℤ 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ Hℤ 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(Hℤ 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. 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 b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (Hℤ 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ Hℤ 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. "This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → 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 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity 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 √ log n.

AB - Given k ∈ N, the k'th discrete Heisenberg group, denoted Hℤ 2k+1, 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 Hℤ 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ Hℤ 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(Hℤ 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. 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 b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (Hℤ 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ Hℤ 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. "This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → 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 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity 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 √ log n.

KW - Approximation algorithms

KW - Heisenberg group

KW - Isoperimetric inequalities

KW - Metric embeddings

KW - Metrics of negative type

KW - Semidefinite programming

KW - Sparsest Cut Problem

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