Expansion in S L d ( 𝒪 K / I ) , I square-free

Péter P. Varjú

Journal of the European Mathematical Society (2012)

  • Volume: 014, Issue: 1, page 273-305
  • ISSN: 1435-9855

Abstract

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Let S be a fixed symmetric finite subset of S L d ( 𝒪 K ) that generates a Zariski dense subgroup of S L d ( 𝒪 K ) when we consider it as an algebraic group over m a t h b b Q by restriction of scalars. We prove that the Cayley graphs of S L d ( 𝒪 K / I ) with respect to the projections of S is an expander family if I ranges over square-free ideals of 𝒪 K if d = 2 and K is an arbitrary numberfield, or if d = 3 and K = ℚ .

How to cite

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Varjú, Péter P.. "Expansion in $SL_d(\mathcal {O}_K/I)$, $I$ square-free." Journal of the European Mathematical Society 014.1 (2012): 273-305. <http://eudml.org/doc/277680>.

@article{Varjú2012,
abstract = {Let $S$ be a fixed symmetric finite subset of $SL_d(\mathcal \{O\}_K)$ that generates a Zariski dense subgroup of $SL_d(\mathcal \{O\}_K)$ when we consider it as an algebraic group over $mathbb Q$ by restriction of scalars. We prove that the Cayley graphs of $SL_d(\mathcal \{O\}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of $\mathcal \{O\}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb \{Q\}$.},
author = {Varjú, Péter P.},
journal = {Journal of the European Mathematical Society},
keywords = {expanders; property tau; Cayley graphs; random walks on groups; affine sieve; expanders; property tau; Cayley graphs; random walks on groups; affine sieves; symmetric generating sets; algebraic groups over number fields},
language = {eng},
number = {1},
pages = {273-305},
publisher = {European Mathematical Society Publishing House},
title = {Expansion in $SL_d(\mathcal \{O\}_K/I)$, $I$ square-free},
url = {http://eudml.org/doc/277680},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Varjú, Péter P.
TI - Expansion in $SL_d(\mathcal {O}_K/I)$, $I$ square-free
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 1
SP - 273
EP - 305
AB - Let $S$ be a fixed symmetric finite subset of $SL_d(\mathcal {O}_K)$ that generates a Zariski dense subgroup of $SL_d(\mathcal {O}_K)$ when we consider it as an algebraic group over $mathbb Q$ by restriction of scalars. We prove that the Cayley graphs of $SL_d(\mathcal {O}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of $\mathcal {O}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb {Q}$.
LA - eng
KW - expanders; property tau; Cayley graphs; random walks on groups; affine sieve; expanders; property tau; Cayley graphs; random walks on groups; affine sieves; symmetric generating sets; algebraic groups over number fields
UR - http://eudml.org/doc/277680
ER -

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