Expansion and random walks in $\mathrm{SL}_d(\mathbb{Z}/p^n\mathbb{ℤ})$: I

Jean Bourgain; Alex Gamburd

doi:10.4171/jems/137

JournalsjemsVol. 10, No. 4pp. 987–1011

Expansion and random walks in $SL_{d} (Z / p^{n} Z)$ : I

Jean Bourgain
Institute for Advanced Study, Princeton, United States
Alex Gamburd
University of California at Santa Cruz, United States

$Expansion and random walks in $\mathrm{SL}_d(\mathbb{Z}/p^n\mathbb{ℤ})$: I cover$

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Abstract

We prove that Cayley graphs of $SL_{2} (Z / p^{n} Z)$ are expanders with respect to the projection of any fixed elements in $SL_{2} (Z)$ generating a Zariski-dense subgroup.

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Jean Bourgain, Alex Gamburd, Expansion and random walks in $SL_{d} (Z / p^{n} Z)$ : I. J. Eur. Math. Soc. 10 (2008), no. 4, pp. 987–1011

DOI 10.4171/JEMS/137