Spectral gap in the group of affine transformations over prime fields

Elon Lindenstrauss; Péter P. Varjú

doi:10.5802/afst.1518

Elon Lindenstrauss ¹; Péter P. Varjú ²

¹ The Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
² University of Cambridge, DPMMS, Wilberforce Road, Cambridge, CB3 0WA, UK

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 969-993.

Abstract
Abstract

We study random walks on the groups $𝔽_{p}^{d} ⋊ {SL}_{d} (𝔽_{p})$ . We estimate the spectral gap in terms of the spectral gap of the projection to the linear part ${SL}_{d} (𝔽_{p})$ . This problem is motivated by an analogue in the group $ℝ^{d} ⋊ SO (d)$ , which have application to smoothness of self-similar measures.

Nous étudions les marches aléatoires sur les groupes $𝔽_{p}^{d} ⋊ {SL}_{d} (𝔽_{p})$ . Nous estimons le trou spectral en fonction du trou spectral de la projection sur la partie linéaire ${SL}_{d} (𝔽_{p})$ . Ce problème est motivé par son analogue dans le groupe $ℝ^{d} ⋊ SO (d)$ , qui a des applications à la régularité des mesures auto-similaires.

Published online: 2016-11-13

MR Zbl

DOI: 10.5802/afst.1518

Author's affiliations:

Elon Lindenstrauss ¹; Péter P. Varjú ²

¹ The Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
² University of Cambridge, DPMMS, Wilberforce Road, Cambridge, CB3 0WA, UK

@article{AFST_2016_6_25_5_969_0,
     author = {Elon Lindenstrauss and P\'eter P. Varj\'u},
     title = {Spectral gap in the group of affine transformations over prime fields},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {969--993},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {5},
     year = {2016},
     doi = {10.5802/afst.1518},
     zbl = {1384.60018},
     mrnumber = {3582116},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1518/}
}

TY  - JOUR
AU  - Elon Lindenstrauss
AU  - Péter P. Varjú
TI  - Spectral gap in the group of affine transformations over prime fields
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2016
SP  - 969
EP  - 993
VL  - 25
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1518/
DO  - 10.5802/afst.1518
LA  - en
ID  - AFST_2016_6_25_5_969_0
ER  -

%0 Journal Article
%A Elon Lindenstrauss
%A Péter P. Varjú
%T Spectral gap in the group of affine transformations over prime fields
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2016
%P 969-993
%V 25
%N 5
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1518/
%R 10.5802/afst.1518
%G en
%F AFST_2016_6_25_5_969_0

Elon Lindenstrauss; Péter P. Varjú. Spectral gap in the group of affine transformations over prime fields. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 969-993. doi : 10.5802/afst.1518. https://afst.centre-mersenne.org/articles/10.5802/afst.1518/

References
Cited by

[1] Bader (U.), Furman (A.), Gelander (T.), and Monod (N.).— Property $(T)$ and rigidity for actions on Banach spaces, Acta Math. 198, no. 1, p. 57-105 (2007). | DOI | MR | Zbl

[2] Bourgain (J.) and Gamburd (A.).— Uniform expansion bounds for Cayley graphs of ${SL}_{2} (𝔽_{p})$ , Ann. of Math. (2) 167, no. 2, p. 625-642 (2008). | DOI

[3] Bourgain (J.) and Gamburd (A.).— Expansion and random walks in ${SL}_{d} (ℤ / p^{n} ℤ)$ . II, J. Eur. Math. Soc. (JEMS) 11, no. 5, p. 1057-1103 (2009). With an appendix by Bourgain. | DOI | MR | Zbl

[4] Breuillard (E.) and Gamburd (A.).— Strong uniform expansion in $S L (2, p)$ , Geom. Funct. Anal. 20, no. 5, p. 1201-1209 (2010). | DOI | MR | Zbl

[5] Breuillard (E.), Green (B.), and Tao (T.).— Approximate subgroups of linear groups, Geom. Funct. Anal. 21, no. 4, p. 774-819 (2011). | DOI | MR | Zbl

[6] Benyamini (Y.) and Lindenstrauss (J.).— Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. | DOI | Zbl

[7] Gowers (W. T.).— Quasirandom groups, Combin. Probab. Comput. 17, no. 3, p. 363-387 (2008). | DOI | MR | Zbl

[8] Helfgott (H. A.).— Growth and generation in ${SL}_{2} (ℤ / p ℤ)$ , Ann. of Math. (2) 167, no. 2, p. 601-623 (2008). | DOI | MR | Zbl

[9] Helfgott (H. A.).— Growth in ${SL}_{3} (ℤ / p ℤ)$ , J. Eur. Math. Soc. (JEMS) 13, no. 3, p. 761-851(2011). | DOI | Zbl

[10] Kowalski (E.).— Crible en expansion, Astérisque 348 (2012).

[11] Kowalski (E.).— Explicit growth and expansion for ${SL}_{2}$ , Int. Math. Res. Not. IMRN 24, p. 5645-5708 (2013). | DOI | MR | Zbl

[12] Landazuri (V.) and Seitz (G. M.).— On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32, p. 418-443 (1974). | DOI | MR | Zbl

[13] Lubotzky (A.).— Expander graphs in pure and applied mathematics, Bull. Amer. Math. Soc. (N.S.) 49, no. 1, p. 113-162 (2012). | DOI | MR | Zbl

[14] Lindenstrauss (E.) and Varju (P. P.).— Spectral gap in the group of affine transformations over prime fields, arXiv preprint arXiv:1409.3564v1 (2014). 26pp. | DOI | MR

[15] Lindenstrauss (E.) and Varju (P. P.).— Random walks in the group of Euclidean isometries and self-similar measures. Duke Math. J. 165, no. 6, p. 1061-1127 (2016). | DOI | MR | Zbl

[16] Lindenstrauss (E.) and Varju (P. P.).— Lectures on dynamical aspects of arithmetic combinatorics. Work in progress.

[17] Lubotzky (A.) and Weiss (B.).— Groups and expanders, Expanding graphs (Princeton, NJ, 1992), p. 95-109 (1993). | DOI

[18] Nikolov (N.) and Pyber (L.).— Product decompositions of quasirandom groups and a Jordan type theorem, J. Eur. Math. Soc. (JEMS) 13, no. 4, p. 1063-1077 (2011). | DOI | MR | Zbl

[19] Pyber (L.) and Szabó (E.).— Growth in finite simple groups of Lie type of bounded rank (2010).

[20] Salehi Golsefidy (A.) and Varju (P. P.).— Expansion in perfect groups, Geom. Funct. Anal. 22, no. 6, p. 1832-1891 (2012). | DOI | MR | Zbl

[21] Sarnak (P.) and Xue (X. X.).— Bounds for multiplicities of automorphic representations, Duke Math. J. 64, no. 1, p. 207-227 (1991). (92h:22026) | DOI | MR | Zbl

[22] Tao (T.).— Product set estimates for non-commutative groups, Combinatorica 28, no. 5, p. 547-594 (2008). | DOI | MR | Zbl

[23] Varju (P. P.).— Expansion in $S L_{d} (𝒪_{K} / I)$ , $I$ square-free, J. Eur. Math. Soc. (JEMS) 14, no. 1, p. 273-305 (2012). | DOI | Zbl

[24] Varju (P. P.).— Random walks in Euclidean space. Ann. of Math. (2), to appear. | DOI | MR | Zbl

Cited by Sources: