Symmetric groups and expanders

Electronic Research Announcements of the American Mathematical Society

Volumn 11, Issue 6, 2005, Pages 47-56

  Kassabov, Martin ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

Alternating groups; Expanders; Kazhdan constants; Property T; Random permutations; Symmetric groups

Indexed keywords

EID: 33646443818     PISSN: None     EISSN: 10796762     Source Type: Journal    
DOI: 10.1090/S1079-6762-05-00146-0     Document Type: Article

References (29)

1. M. Abért, Symmetric groups as products of abelian subgroups, Bull. London Math. Soc. 34 (2002), no. 4, 451–456. MR1897424 (2002m:20006)
2. N. Alon, A. Lubotzky, and A. Wigderson, Semi-direct product in groups and zig-zag product in graphs: connections and applications (extended abstract), 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, 2001, pp. 630–637. MR1948752
3. N. Alon and Y. Roichman, Random Cayley graphs and expanders, Random Structures Algorithms 5 (1994), no. 2, 271–284. MR1262979 (94k:05132)
4. L. Babai, G. Hetyei, W. M. Kantor, A. Lubotzky, and ´A. Seress, On the diameter of finite groups, 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), IEEE Comput. Soc. Press, Los Alamitos, CA, 1990, pp. 857–865. MR1150735
5. L. Babai, W. M. Kantor, and A. Lubotzky, Small-diameter Cayley graphs for finite simple groups, European J. Combin. 10 (1989), no. 6, 507–522. MR1022771 (91a:20038)
6. M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math. 413 (1991), 36–67. MR1089795 (92c:22013)
7. R. Gilman, Finite quotients of the automorphism group of a free group, Canad. J. Math. 29 (1977), no. 3, 541–551. MR0435226 (55:8186)
8. M. Kassabov, Kazhdan constants for SLn(Z), arXiv:math.GR/0311487.
9. M. Kassabov, Symmetric groups and expanders, in preparation.
10. M. Kassabov, Universal lattices and unbounded rank expanders, arXiv:math.GR/0502237.
11. M. Kassabov and N. Nikolov, Finite simple groups and expanders, in preparation.
12. M. Kassabov and T. R. Riley, Diameters of Cayley graphs of SLn(Z/kZ), arXiv:math.GR/0502221.
13. D. A. Kazhdan, On the connection of the dual space of a group with the structure of its closed subgroups, Funktsional. Anal. i Prilozhen. 1 (1967), 71–74. MR0209390 (35:288)
14. Z. Landau and A. Russell, Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem, Electron. J. Combin. 11 (2004), no. 1, Research Paper 62, 6 pp. (electronic). MR2097328
15. P. Loh and L. Schulman, Improved expansion of random cayley graphs, Disc. Math. and Theor. Comp. Sci. 6 (2004), 523–528. MR2097328
16. A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277. MR963118 (89m:05099)
17. A. Lubotzky and B. Weiss, Groups and expanders, Expanding graphs (Princeton, NJ, 1992), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 10, Amer. Math. Soc., Providence, RI, 1993, pp. 95–109. MR1235570 (95b:05097)
18. A. Lubotzky, private communication.
19. A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser Verlag, Basel, 1994. MR1308046 (96g:22018)
20. A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks, Surveys in combinatorics, 1995 (Stirling), London Math. Soc. Lecture Note Ser., vol. 218, Cambridge Univ. Press, Cambridge, 1995, pp. 155–189. MR1358635 (96k:05081)
21. A. Lubotzky and I. Pak, The product replacement algorithm and Kazhdan’s property (T), J. Amer. Math. Soc. 14 (2001), no. 2, 347–363 (electronic). MR1815215 (2003d:60012)
22. A. Lubotzky and A. Żuk, On property τ, preprint. http://www.ma.huji.ac.il/alexlub/BOOKS/On%20property/On%20property.pdf
23. G. A. Margulis, Explicit constructions of expanders, Problemy Peredachi Informatsii 9 (1973), no. 4, 71–80. MR0484767 (58:4643)
24. N. Nikolov, private communication.
25. O. Reingold, S. Vadhan, and A. Wigderson, Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors (extended abstract), 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, pp. 3–13. MR1931799
26. Y. Roichman, Upper bound on the characters of the symmetric groups, Invent. Math. 125 (1996), no. 3, 451–485. MR1400314 (97e:20014)
27. Y. Roichman, Expansion properties of Cayley graphs of the alternating groups, J. Combin. Theory Ser. A 79 (1997), no. 2, 281–297. MR1462559 (98g:05070)
28. E. Rozenman, Aner Shalev, and Avi Wigderson, A new family of Cayley expanders, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 445–454 (electronic), ACM, New York, 2004. MR2121630
29. Y. Shalom, Bounded generation and Kazhdan’s property (T), Inst. Hautes Études Sci. Publ. Math. no. 90 (1999), 145–168 (2001). MR1813225 (2001m:22030)