Show Summary Details
More options …

Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

2 Issues per year

CiteScore 2017: 0.32

SCImago Journal Rank (SJR) 2017: 0.208
Source Normalized Impact per Paper (SNIP) 2017: 0.322

Mathematical Citation Quotient (MCQ) 2017: 0.32

Online
ISSN
1869-6104
See all formats and pricing
More options …
Volume 10, Issue 1

Certifying numerical estimates of spectral gaps

Marek Kaluba
/ Piotr W. Nowak
Published Online: 2018-04-20 | DOI: https://doi.org/10.1515/gcc-2018-0004

Abstract

We establish a lower bound on the spectral gap of the Laplace operator on special linear groups using conic optimisation. In particular, this provides a constructive (but computer assisted) proof that these groups have the Kazhdan property (T). Software for such optimisation for other finitely presented groups is provided.

MSC 2010: 16S34; 20C07; 20C40

References

• [1]

B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T), New Math. Monogr. 11, Cambridge University Press, Cambridge, 2008. Google Scholar

• [2]

J. Bezanson, A. Edelman, S. Karpinski and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev. 59 (2017), no. 1, 65–98.

• [3]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. Google Scholar

• [4]

M. Burger, Kazhdan constants for $\mathrm{SL}\left(3,𝐙\right)$, J. Reine Angew. Math. 413 (1991), 36–67. Google Scholar

• [5]

I. Dunning, J. Huchette and M. Lubin, JuMP: A modeling language for mathematical optimization, SIAM Rev. 59 (2017), no. 2, 295–320.

• [6]

K. Fujiwara and Y. Kabaya, Computing Kazhdan constants by semidefinite programming, preprint (2017), https://arxiv.org/abs/1703.04555.

• [7]

U. Hadad, Uniform Kazhdan constant for some families of linear groups, J. Algebra 318 (2007), no. 2, 607–618.

• [8]

B. He and X. Yuan, On the $O\left(1/n\right)$ convergence rate of the Douglas–Rachford alternating direction method, SIAM J. Numer. Anal. 50 (2012), no. 2, 700–709.

• [9]

M. Kassabov, Kazhdan constants for ${\mathrm{SL}}_{n}\left(ℤ\right)$, Internat. J. Algebra Comput. 15 (2005), no. 5–6, 971–995. Google Scholar

• [10]

T. Netzer, Real algebraic geometry and its applications, preprint (2016), https://arxiv.org/abs/1606.07284.

• [11]

T. Netzer and A. Thom, Real closed separation theorems and applications to group algebras, Pacific J. Math. 263 (2013), no. 2, 435–452.

• [12]

T. Netzer and A. Thom, Kazhdan’s property (T) via semidefinite optimization, Exp. Math. 24 (2015), no. 3, 371–374.

• [13]

B. O’Donoghue, E. Chu, N. Parikh and S. Boyd, Conic optimization via operator splitting and homogeneous self-dual embedding, J. Optim. Theory Appl. 169 (2016), no. 3, 1042–1068.

• [14]

N. Ozawa, Noncommutative real algebraic geometry of Kazhdan’s property (T), J. Inst. Math. Jussieu 15 (2016), no. 1, 85–90.

• [15]

K. Schmüdgen, Noncommutative real algebraic geometry – some basic concepts and first ideas, Emerging Applications of Algebraic Geometry, Springer, New York (2009), 325–350. Google Scholar

• [16]

Y. Shalom, Bounded generation and Kazhdan’s property (T), Publ. Math. Inst. Hautes Études Sci. (1999), no. 90, 145–168. Google Scholar

• [17]

W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, 2011. Google Scholar

• [18]

L. Tunçel, Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization, Fields Inst. Monogr. 27, American Mathematical Society, Providence, 2010. Google Scholar

• [19]

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38 (1996), no. 1, 49–95.

• [20]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.6, 2016.

Published Online: 2018-04-20

Published in Print: 2018-05-01

Funding Source: Narodowe Centrum Nauki

Award identifier / Grant number: 2015/19/B/ST1/01458

Funding Source: H2020 European Research Council

Award identifier / Grant number: 677120-INDEX

The first author has been partially supported by the National Science Centre, under grant number 2015/19/B/ST1/01458. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 677120-INDEX).

Citation Information: Groups Complexity Cryptology, Volume 10, Issue 1, Pages 33–41, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.