A Cheeger-Buser-type inequality on CW complexes

Document Type : Ischia Group Theory 2020/2021


Section de matématiques, University of Geneva


We extend the definition of boundary expansion to CW complexes and prove a Cheeger-Buser-type relation between the spectral gap of the Laplacian and the boundary expansion of an orientable CW complex.


Main Subjects

[1] B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge
University Press, Cambridge, 2008.
[2] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to
Salomon Bochner, 1969), Princeton Univ. Press, Princeton, N. J., 1970 195–199.
[3] D. Dotterrer, The (co)isoperimetric problem in (random) polyhedra, Thesis (Ph.D.)–University of Toronto (Canada).
ProQuest LLC, Ann Arbor, MI, 2013 90 pp.
[4] K. Golubev and O. Parzanchevski, Spectrum and combinatorics of two-dimensional Ramanujan complexes, Israel
J. Math., 230 (2019) 583–612.
[5] M. Gromov, Singularities, expanders and topology of maps, Part 2: From combinatorics to topology via algebraic
isoperimetry, Geom. Funct. Anal., 20 (2010) 416–526.
[6] A. Gundert and M. Szedlák, Higher dimensional discrete Cheeger inequalities, J. Comput. Geom., 6 (2015) 54–71.
[7] A. Gundert and U. Wagner, On Laplacians of random complexes, Computational geometry (SCG’12), ACM, New
York, 2012 151–160.
[8] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
[9] S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.), 43
(2006) 439–561.
[10] N. Linial and R. Meshulam, Homological connectivity of random 2-complexes, Combinatorica, 26 (2006) 475–487.
[11] A. Lubotzky, Expander graphs in pure and applied mathematics, Bull. Amer. Math. Soc. (N.S.), 49 (2012) 113–162.
[12] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete,
Band 89, Springer-Verlag, Berlin-New York, 1977.
[13] W. S. Massey, Singular homology theory, Graduate Texts in Mathematics, 70, Springer-Verlag, New York-Berlin,
[14] R. Meshulam and N. Wallach, Homological connectivity of random k-dimensional complexes, Random Structures
Algorithms, 34 (2009) 408–417.
[15] O. Parzanchevski, High Dimensional Expanders, Ph. D. thesis, Hebrew University of Jerusalem, 2013.
[16] O. Parzanchevski, R. Rosenthal and R. J. Tessler, Isoperimetric inequalities in simplicial complexes, Combinatorica,
36 (2016) 195–227.
[17] J. Steenbergen, C. Klivans and S. Mukherjee, A Cheeger-type inequality on simplicial complexes, Adv. in Appl.
Math., 56 (2014) 56–77.
Volume 12, Issue 3 - Serial Number 3
Proceedings of the Ischia Group Theory (2020/2021) - Part 3
September 2023
Pages 197-204
  • Receive Date: 31 December 2021
  • Revise Date: 21 June 2022
  • Accept Date: 23 June 2022
  • Published Online: 01 September 2023