Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps
We introduce the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacia...
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| Published in | Discrete & computational geometry Vol. 59; no. 2; pp. 293 - 330 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.03.2018
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0179-5376 1432-0444 |
| DOI | 10.1007/s00454-017-9948-x |
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| Abstract | We introduce the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex. |
|---|---|
| AbstractList | We introduce the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex. |
| Author | Oppenheim, Izhar |
| Author_xml | – sequence: 1 givenname: Izhar orcidid: 0000-0002-1849-1851 surname: Oppenheim fullname: Oppenheim, Izhar email: izharo@bgu.ac.il organization: Department of Mathematics, Ben-Gurion University of the Negev |
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| CitedBy_id | crossref_primary_10_1007_s00454_019_00117_7 crossref_primary_10_1016_j_ipl_2022_106268 crossref_primary_10_1016_j_aim_2025_110117 crossref_primary_10_1137_20M1383689 crossref_primary_10_2140_apde_2023_16_861 crossref_primary_10_1007_s11856_023_2511_y crossref_primary_10_1007_s11856_023_2521_9 crossref_primary_10_1090_jams_1019 crossref_primary_10_1093_imrn_rnab126 crossref_primary_10_1016_j_ejc_2023_103696 crossref_primary_10_1016_j_ejc_2025_104131 crossref_primary_10_1007_s11856_023_2480_1 crossref_primary_10_1137_21M1443340 crossref_primary_10_1145_3631354 crossref_primary_10_1137_22M1474734 crossref_primary_10_1007_s00493_024_00084_5 crossref_primary_10_1137_20M1367696 crossref_primary_10_1007_s00454_018_9991_2 crossref_primary_10_1007_s41468_023_00158_1 crossref_primary_10_1145_3531008 crossref_primary_10_1007_s00493_019_3847_0 |
| Cites_doi | 10.1002/rsa.20238 10.2307/1970829 10.1007/s00039-010-0073-8 10.1016/j.aim.2013.05.007 10.1007/BFb0066362 10.1090/S0273-0979-2011-01359-3 10.1017/S0963548317000116 10.1007/s00493-014-3002-x 10.4171/GGD/346 10.1007/s000390050022 10.1007/s00493-006-0027-9 10.1007/s11537-014-1265-z |
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| Keywords | Graph Laplacian Spectral gap Secondary 05A20 05C81 High dimensional expanders Simplicial complexes Primary 05E45 |
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| References_xml | – reference: LinialNMeshulamRHomological connectivity of random 2-complexesCombinatorica2006264475487226085010.1007/s00493-006-0027-91121.55013 – reference: GromovMSingularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetryGeom. Funct. Anal.2010202416526267128410.1007/s00039-010-0073-81251.05039 – reference: Borel, A.: Cohomologie de certains groupes discretes et laplacien p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-adique (d’après H. Garland). In: Séminaire Bourbaki, 26e année (1973/1974), Exp. No. 437. Lecture Notes in Mathematics, Vol. 431, pp. 12–35. Springer, Berlin (1975) – reference: HorakDJostJSpectra of combinatorial Laplace operators on simplicial complexesAdv. Math.2013244303336307787410.1016/j.aim.2013.05.0071290.05103 – reference: ParzanchevskiOMixing in high-dimensional expandersComb. Probab. Comput.2017265746761368198010.1017/S09635483170001161371.05329 – reference: LubotzkyARamanujan complexes and high dimensional expandersJpn. J. Math.201492137169325861710.1007/s11537-014-1265-z1302.05095 – reference: LubotzkyAMeshulamRMozesSExpansion of building-like complexesGroups Geom. Dyn.2016101155175346033410.4171/GGD/34606558482 – reference: Ballmann, W., Świątkowski, J.: On L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-cohomology and property (T) for automorphism groups of polyhedral cell complexes. Geom. Funct. Anal. 7(4), 615–645 (1997) – reference: ParzanchevskiORosenthalRTesslerRJIsoperimetric inequalities in simplicial complexesCombinatorica2016362195227351688410.1007/s00493-014-3002-x06766139 – reference: LubotzkyAExpander graphs in pure and applied mathematicsBull. Am. Math. Soc. (N.S.)2012491113162286901010.1090/S0273-0979-2011-01359-31232.05194 – reference: MeshulamRWallachNHomological connectivity of random k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-dimensional complexesRandom Struct. Algorithms2009343408417250440510.1002/rsa.202381177.55011 – reference: GarlandHp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-Adic curvature and the cohomology of discrete subgroups of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-adic groupsAnn. Math.197397337542332018010.2307/19708290262.22010 – volume: 34 start-page: 408 issue: 3 year: 2009 ident: 9948_CR10 publication-title: Random Struct. Algorithms doi: 10.1002/rsa.20238 – volume: 97 start-page: 375 issue: 3 year: 1973 ident: 9948_CR3 publication-title: Ann. Math. doi: 10.2307/1970829 – volume: 20 start-page: 416 issue: 2 year: 2010 ident: 9948_CR4 publication-title: Geom. Funct. Anal. doi: 10.1007/s00039-010-0073-8 – volume: 244 start-page: 303 year: 2013 ident: 9948_CR5 publication-title: Adv. Math. doi: 10.1016/j.aim.2013.05.007 – ident: 9948_CR2 doi: 10.1007/BFb0066362 – volume: 49 start-page: 113 issue: 1 year: 2012 ident: 9948_CR7 publication-title: Bull. Am. Math. Soc. (N.S.) doi: 10.1090/S0273-0979-2011-01359-3 – volume: 26 start-page: 746 issue: 5 year: 2017 ident: 9948_CR11 publication-title: Comb. Probab. Comput. doi: 10.1017/S0963548317000116 – volume: 36 start-page: 195 issue: 2 year: 2016 ident: 9948_CR12 publication-title: Combinatorica doi: 10.1007/s00493-014-3002-x – volume: 10 start-page: 155 issue: 1 year: 2016 ident: 9948_CR9 publication-title: Groups Geom. Dyn. doi: 10.4171/GGD/346 – ident: 9948_CR1 doi: 10.1007/s000390050022 – volume: 26 start-page: 475 issue: 4 year: 2006 ident: 9948_CR6 publication-title: Combinatorica doi: 10.1007/s00493-006-0027-9 – volume: 9 start-page: 137 issue: 2 year: 2014 ident: 9948_CR8 publication-title: Jpn. J. Math. doi: 10.1007/s11537-014-1265-z |
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| Title | Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps |
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