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Licensed Unlicensed Requires Authentication Published by De Gruyter October 9, 2021

Logarithmic diameter bounds for some Cayley graphs

Lam Pham and Xin Zhang
From the journal Journal of Group Theory

Abstract

Let S GL n ( Z ) be a finite symmetric set. We show that if the Zariski closure of Γ = S is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay ( Γ / Γ ( q ) , π q ( S ) ) is O ( log q ) , where 𝑞 is an arbitrary positive integer, π q : Γ Γ / Γ ( q ) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.

Acknowledgements

We thank the anonymous referee for the numerous helpful corrections and suggestions on this work.

  1. Communicated by: Adrian Ioana

References

[1] J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, J. Amer. Math. Soc. 24 (2011), no. 1, 231–280. 10.1090/S0894-0347-2010-00674-1Search in Google Scholar

[2] J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL 2 ( F p ) , Ann. of Math. (2) 167 (2008), no. 2, 625–642. 10.4007/annals.2008.167.625Search in Google Scholar

[3] J. Bourgain, A. Gamburd and P. Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), no. 3, 559–644. 10.1007/s00222-009-0225-3Search in Google Scholar

[4] J. Bourgain and A. Kontorovich, On representations of integers in thin subgroups of SL 2 ( Z ) , Geom. Funct. Anal. 20 (2010), no. 5, 1144–1174. 10.1007/s00039-010-0093-4Search in Google Scholar

[5] J. Bourgain and A. Kontorovich, On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), no. 3, 589–650. 10.1007/s00222-013-0475-ySearch in Google Scholar

[6] J. Bourgain and A. Kontorovich, On Zaremba’s conjecture, Ann. of Math. (2) 180 (2014), no. 1, 137–196. 10.4007/annals.2014.180.1.3Search in Google Scholar

[7] J. Bourgain and P. P. Varjú, Expansion in S L d ( Z / q Z ) , q arbitrary, Invent. Math. 188 (2012), no. 1, 151–173. 10.1007/s00222-011-0345-4Search in Google Scholar

[8] E. Breuillard, B. Green and T. Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), no. 4, 774–819. 10.1007/s00039-011-0122-ySearch in Google Scholar

[9] E. Fuchs, K. E. Stange and X. Zhang, Local-global principles in circle packings, Compos. Math. 155 (2019), no. 6, 1118–1170. 10.1112/S0010437X19007139Search in Google Scholar

[10] A. S. Golsefidy and P. P. Varjú, Expansion in perfect groups, Geom. Funct. Anal. 22 (2012), no. 6, 1832–1891. 10.1007/s00039-012-0190-7Search in Google Scholar

[11] W. He and N. de Saxcé, Linear random walks on the torus, preprint (2019), https://arxiv.org/abs/1910.13421. 10.1215/00127094-2021-0045Search in Google Scholar

[12] H. A. Helfgott, Growth and generation in SL 2 ( Z / p Z ) , Ann. of Math. (2) 167 (2008), no. 2, 601–623. 10.4007/annals.2008.167.601Search in Google Scholar

[13] M. V. Nori, On subgroups of GL n ( F p ) , Invent. Math. 88 (1987), no. 2, 257–275. 10.1007/BF01388909Search in Google Scholar

[14] L. Pyber and E. Szabó, Growth in finite simple groups of Lie type, J. Amer. Math. Soc. 29 (2016), no. 1, 95–146. 10.1090/S0894-0347-2014-00821-3Search in Google Scholar

[15] A. Salehi Golsefidy, Super-approximation, II: The 𝑝-adic case and the case of bounded powers of square-free integers, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 7, 2163–2232. 10.4171/JEMS/883Search in Google Scholar

[16] A. Salehi Golsefidy and P. Sarnak, The affine sieve, J. Amer. Math. Soc. 26 (2013), no. 4, 1085–1105. 10.1090/S0894-0347-2013-00764-XSearch in Google Scholar

[17] X. Zhang, On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math. 737 (2018), 71–110. 10.1515/crelle-2015-0042Search in Google Scholar

[18] X. Zhang, On representation of integers from thin subgroups of SL ( 2 , Z ) with parabolics, Int. Math. Res. Not. IMRN 2020 (2020), no. 18, 5611–5629. 10.1093/imrn/rny177Search in Google Scholar

Received: 2020-07-16
Revised: 2021-09-15
Published Online: 2021-10-09
Published in Print: 2022-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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