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- Superstrong_approximation abstract "Superstrong approximation is a generalisation of strong approximation in algebraic groups G, to provide \"spectral gap\" results. The spectrum in question is that of the Laplacian matrix associated to a family of quotients of a discrete group Γ; and the gap is that between the first and second eigenvalues (normalisation so that the first eigenvalue corresponds to constant functions as eigenvectors). Here Γ is a subgroup of the rational points of G, but need not be a lattice: it may be a so-called thin group. The \"gap\" in question is a lower bound (absolute constant) for the difference of those eigenvalues.A consequence and equivalent of this property, potentially holding for Zariski dense subgroups Γ of the special linear group over the integers, and in more general classes of algebraic groups G, is that the sequence of Cayley graphs for reductions Γp modulo prime numbers p, with respect to any fixed set S in Γ that is a symmetric set and generating set, is an expander family.In this context \"strong approximation\" is the statement that S when reduced generates the full group of points of G over the prime fields with p elements, when p is large enough. It is equivalent to the Cayley graphs being connected (when p is large enough), or that the locally constant functions on these graphs are constant, so that the eigenspace for the first eigenvalue is one-dimensional. Superstrong approximation therefore is a concrete quantitative improvement on these statements.".
- Superstrong_approximation wikiPageExternalLink index.html.
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- Superstrong_approximation wikiPageRevisionID "639322824".
- Superstrong_approximation wikiPageWikiLink Alexander_Lubotzky.
- Superstrong_approximation wikiPageWikiLink Approximate_subgroup.
- Superstrong_approximation wikiPageWikiLink Approximation_in_algebraic_groups.
- Superstrong_approximation wikiPageWikiLink Category:Algebraic_groups.
- Superstrong_approximation wikiPageWikiLink Category:Cayley_graphs.
- Superstrong_approximation wikiPageWikiLink Category:Spectral_theory.
- Superstrong_approximation wikiPageWikiLink Cayley_graph.
- Superstrong_approximation wikiPageWikiLink Expander_graph.
- Superstrong_approximation wikiPageWikiLink Generator_(mathematics).
- Superstrong_approximation wikiPageWikiLink Growth_rate_(group_theory).
- Superstrong_approximation wikiPageWikiLink Kazhdans_property_(T).
- Superstrong_approximation wikiPageWikiLink Laplacian_matrix.
- Superstrong_approximation wikiPageWikiLink Lattice_(discrete_subgroup).
- Superstrong_approximation wikiPageWikiLink Special_linear_group.
- Superstrong_approximation wikiPageWikiLink Symmetric_set.
- Superstrong_approximation wikiPageWikiLink Thin_group_(algebraic_group_theory).
- Superstrong_approximation wikiPageWikiLink Zariski_topology.
- Superstrong_approximation wikiPageWikiLinkText "Superstrong approximation".
- Superstrong_approximation wikiPageUsesTemplate Template:Citation.
- Superstrong_approximation wikiPageUsesTemplate Template:Reflist.
- Superstrong_approximation subject Category:Algebraic_groups.
- Superstrong_approximation subject Category:Cayley_graphs.
- Superstrong_approximation subject Category:Spectral_theory.
- Superstrong_approximation hypernym Generalisation.
- Superstrong_approximation comment "Superstrong approximation is a generalisation of strong approximation in algebraic groups G, to provide \"spectral gap\" results. The spectrum in question is that of the Laplacian matrix associated to a family of quotients of a discrete group Γ; and the gap is that between the first and second eigenvalues (normalisation so that the first eigenvalue corresponds to constant functions as eigenvectors).".
- Superstrong_approximation label "Superstrong approximation".
- Superstrong_approximation sameAs Q17103817.
- Superstrong_approximation sameAs m.010pr0jm.
- Superstrong_approximation sameAs Q17103817.
- Superstrong_approximation wasDerivedFrom Superstrong_approximation?oldid=639322824.
- Superstrong_approximation isPrimaryTopicOf Superstrong_approximation.