| dbo: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. (en)
|
| dbo:wikiPageExternalLink
| |
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 3931 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:wikiPageUsesTemplate
| |
| dcterms:subject
| |
| gold:hypernym
| |
| rdf:type
| |
| rdfs: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). 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. (en)
|
| rdfs:label
|
- Superstrong approximation (en)
|
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageDisambiguates
of | |
| is dbo:wikiPageRedirects
of | |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
