Strong approximation in the Apollonian group

Elena Fuchs

Research output: Contribution to journalArticlepeer-review

Abstract

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group OQ(Z{double-struck}) fixing the Descartes quadratic form Q. For nonzero v∈Z{double-struck}4 satisfying Q(v)=0, the orbits Pv=Av correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits Pv mod any integer d>1. We show that this reduction has a multiplicative structure, and that mod primes p≥5 it is the full cone of integer solutions to Q(v)≡0 for v≠0. This analysis is an essential ingredient in applications of the affine linear sieve as developed by Bourgain, Gamburd and Sarnak.

Original languageEnglish (US)
Pages (from-to)2282-2302
Number of pages21
JournalJournal of Number Theory
Volume131
Issue number12
DOIs
StatePublished - Dec 2011
Externally publishedYes

Keywords

  • Affine sieve
  • Apollonian circle packings
  • Congruence obstructions
  • Local to global

ASJC Scopus subject areas

  • Algebra and Number Theory

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