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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Superstrong approximation</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="This article exist only in this language. 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Superstrong approximation</b> is a generalisation of <a href="/wiki/Strong_approximation_in_algebraic_groups" class="mw-redirect" title="Strong approximation in algebraic groups">strong approximation in algebraic groups</a> <i>G</i>, to provide <a href="/wiki/Spectral_gap" title="Spectral gap">spectral gap</a> results. The spectrum in question is that of the <a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian matrix</a> 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 <i>G</i>, but need not be a <a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">lattice</a>: it may be a so-called <a href="/wiki/Thin_group_(algebraic_group_theory)" title="Thin group (algebraic group theory)">thin group</a>. The "gap" in question is a lower bound (absolute constant) for the difference of those eigenvalues. </p><p>A consequence and equivalent of this property, potentially holding for <a href="/wiki/Zariski_dense" class="mw-redirect" title="Zariski dense">Zariski dense</a> subgroups Γ of the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> over the integers, and in more general classes of algebraic groups <i>G</i>, is that the sequence of <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graphs</a> for reductions Γ<sub><i>p</i></sub> modulo prime numbers <i>p</i>, with respect to any fixed set <i>S</i> in Γ that is a <a href="/wiki/Symmetric_set" title="Symmetric set">symmetric set</a> and <a href="/wiki/Generating_set" class="mw-redirect" title="Generating set">generating set</a>, is an <a href="/wiki/Expander_family" class="mw-redirect" title="Expander family">expander family</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In this context "strong approximation" is the statement that <i>S</i> when reduced generates the full group of points of <i>G</i> over the prime fields with <i>p</i> elements, when <i>p</i> is large enough. It is equivalent to the Cayley graphs being connected (when <i>p</i> 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. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superstrong_approximation&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Property <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d11091add00c31137100b463d09eecb6d13578b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.011ex; height:2.843ex;" alt="{\displaystyle (\tau )}"></span></b> is an analogue in discrete group theory of <a href="/wiki/Kazhdan%27s_property_(T)" title="Kazhdan's property (T)">Kazhdan's property (T)</a>, and was introduced by <a href="/wiki/Alexander_Lubotzky" title="Alexander Lubotzky">Alexander Lubotzky</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For a given family of normal subgroups <i>N</i> of finite index in Γ, one equivalent formulation is that the Cayley graphs of the groups Γ/<i>N</i>, all with respect to a fixed symmetric set of generators <i>S</i>, form an expander family.<sup id="cite_ref-Lubotzky1994_3-0" class="reference"><a href="#cite_note-Lubotzky1994-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Therefore superstrong approximation is a formulation of property <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d11091add00c31137100b463d09eecb6d13578b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.011ex; height:2.843ex;" alt="{\displaystyle (\tau )}"></span>, where the subgroups <i>N</i> are the kernels of reduction modulo large enough primes <i>p</i>. </p><p>The <b>Lubotzky–Weiss conjecture</b> states (for special linear groups and reduction modulo primes) that an expansion result of this kind holds independent of the choice of <i>S</i>. For applications, it is also relevant to have results where the modulus is not restricted to being a prime.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Proofs_of_superstrong_approximation">Proofs of superstrong approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superstrong_approximation&action=edit&section=2" title="Edit section: Proofs of superstrong approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Results on superstrong approximation have been found using techniques on <a href="/w/index.php?title=Approximate_subgroup&action=edit&redlink=1" class="new" title="Approximate subgroup (page does not exist)">approximate subgroups</a>, and <a href="/wiki/Growth_rate_(group_theory)" title="Growth rate (group theory)">growth rate</a> in finite simple groups.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superstrong_approximation&action=edit&section=3" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBreuillardOh2014">Breuillard & Oh 2014</a>, pages x, 343)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLubotzky2005" class="citation journal cs1"><a href="/wiki/Alexander_Lubotzky" title="Alexander Lubotzky">Lubotzky, Alex</a> (2005). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200506/what-is.pdf">"What is ... property <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d11091add00c31137100b463d09eecb6d13578b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.011ex; height:2.843ex;" alt="{\displaystyle (\tau )}"></span>?"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>. <b>52</b> (6): 626–627. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2147485">2147485</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=What+is+...+property+MATH+RENDER+ERROR%3F&rft.volume=52&rft.issue=6&rft.pages=626-627&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2147485%23id-name%3DMR&rft.aulast=Lubotzky&rft.aufirst=Alex&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200506%2Fwhat-is.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperstrong+approximation" class="Z3988"></span></span> </li> <li id="cite_note-Lubotzky1994-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lubotzky1994_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander_Lubotzky1994" class="citation book cs1"><a href="/wiki/Alexander_Lubotzky" title="Alexander Lubotzky">Alexander Lubotzky</a> (1 January 1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aNURlzNuotEC&pg=PA49"><i>Discrete Groups, Expanding Graphs and Invariant Measures</i></a>. Springer. p. 49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-5075-8" title="Special:BookSources/978-3-7643-5075-8"><bdi>978-3-7643-5075-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Groups%2C+Expanding+Graphs+and+Invariant+Measures&rft.pages=49&rft.pub=Springer&rft.date=1994-01-01&rft.isbn=978-3-7643-5075-8&rft.au=Alexander+Lubotzky&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaNURlzNuotEC%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperstrong+approximation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBreuillardOh2014">Breuillard & Oh 2014</a>, pages 3-4)</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">(<a href="#CITEREFBreuillardOh2014">Breuillard & Oh 2014</a>, page xi)</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superstrong_approximation&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBreuillardOh2014" class="citation book cs1">Breuillard, Emmanuel; Oh, Hee, eds. (2014). <a rel="nofollow" class="external text" href="http://library.msri.org/books/Book61/index.html"><i>Thin Groups and Superstrong Approximation</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-03685-7" title="Special:BookSources/978-1-107-03685-7"><bdi>978-1-107-03685-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thin+Groups+and+Superstrong+Approximation&rft.pub=Cambridge+University+Press&rft.date=2014&rft.isbn=978-1-107-03685-7&rft_id=http%3A%2F%2Flibrary.msri.org%2Fbooks%2FBook61%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperstrong+approximation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatthewsVasersteinWeisfeiler1984" class="citation journal cs1">Matthews, C. R.; Vaserstein, L. N.; Weisfeiler, B. (1984). "Congruence properties of Zariski-dense subgroups. I.". <i>Proc. London Math. Soc</i>. Series 3. <b>48</b> (3): 514–532. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs3-48.3.514">10.1112/plms/s3-48.3.514</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0735226">0735226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+London+Math.+Soc.&rft.atitle=Congruence+properties+of+Zariski-dense+subgroups.+I.&rft.volume=48&rft.issue=3&rft.pages=514-532&rft.date=1984&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs3-48.3.514&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0735226%23id-name%3DMR&rft.aulast=Matthews&rft.aufirst=C.+R.&rft.au=Vaserstein%2C+L.+N.&rft.au=Weisfeiler%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperstrong+approximation" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Superstrong_approximation&oldid=1220166173">https://en.wikipedia.org/w/index.php?title=Superstrong_approximation&oldid=1220166173</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Algebraic_groups" title="Category:Algebraic groups">Algebraic groups</a></li><li><a href="/wiki/Category:Cayley_graphs" title="Category:Cayley graphs">Cayley graphs</a></li><li><a href="/wiki/Category:Spectral_theory" title="Category:Spectral theory">Spectral theory</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 22 April 2024, at 05:51<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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