Strong approximation in the Apollonian group
نویسندگان
چکیده
منابع مشابه
Strong Approximation in the Apollonian Group
The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group OQ(Z) fixing the Descartes quadratic form Q. For nonzero v ∈ Z4 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...
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Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)×(center) is an integer vector. This series of papers explain such...
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A Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. Part I shoewed there is a natural group action on Desc...
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Here ZS = Z[1/p | p ∈ S]. In case (b) we can deduce from the Strong Approximation Theorem that ∆1 has many finite images, in particular the groups ∏k i=1 G(Fpi) whenever p1, . . . , pk are distinct primes outside S. Now, for all but finitely many primes p the group G(Fp) is semisimple, in fact it is a perfect central extension of a product of simple groups (of fixed Lie type over Fp). The simpl...
متن کاملApollonian Circle Packings : Geometry and Group Theory
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)×(center) is an integer vector. This series of papers explain such...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2011
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2011.05.010