{"paper":{"title":"Vertical perimeter versus horizontal perimeter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.CA","math.CO","math.FA"],"primary_cat":"math.MG","authors_text":"Assaf Naor, Robert Young","submitted_at":"2017-01-03T10:25:43Z","abstract_excerpt":"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 $\\Omega$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00620","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}