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<title>Totally disconnected group</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Totally disconnected group</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>totally disconnected group</b> is a <a href="Topological_group" title="Topological group">topological group</a> that is <a href="Totally_disconnected" class="mw-redirect" title="Totally disconnected">totally disconnected</a>. Such topological groups are necessarily <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a>.

</p><p>Interest centres on <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> totally disconnected groups (variously referred to as groups of <b>td-type</b>,<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> <a href="Locally_profinite_group" title="Locally profinite group">locally profinite groups</a>,<sup id="cite_ref-BushnellHenniart_2-0" class="reference"><a href="#cite_note-BushnellHenniart-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or <b>t.d. groups</b><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>). The <a href="Compact_space" title="Compact space">compact</a> case has been heavily studied – these are the <a href="Profinite_group" title="Profinite group">profinite groups</a> – but for a long time not much was known about the general case. A theorem of <a href="David_van_Dantzig" title="David van Dantzig">van Dantzig</a><sup id="cite_ref-Dantzig_4-0" class="reference"><a href="#cite_note-Dantzig-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> from the 1930s, stating that every such group contains a compact <a href="Open_set" title="Open set">open</a> <a href="Subgroup" title="Subgroup">subgroup</a>, was all that was known. Then groundbreaking work by <a href="George_A._Willis" title="George A. Willis">George Willis</a> in 1994,<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> opened up the field by showing that every locally compact totally disconnected group contains a so-called <i>tidy</i> subgroup and a special function on its <a href="Automorphism" title="Automorphism">automorphisms</a>, the <i>scale function</i>, giving a quantifiable parameter for the local structure. Advances on the <i>global structure</i> of totally disconnected groups were obtained in 2011 by Caprace and <a href="Nicolas_Monod" title="Nicolas Monod">Monod</a>, with notably a classification of <a href="Characteristically_simple_group" title="Characteristically simple group">characteristically simple groups</a> and of <a href="Noetherian_group" class="mw-redirect" title="Noetherian group">Noetherian groups</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Locally_compact_case">Locally compact case</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Locally_profinite_group" title="Locally profinite group">Locally profinite group</a></div>

<p>In a locally compact, totally disconnected group, every <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of the identity contains a compact open subgroup. Conversely, if a group is such that the identity has a <a href="Neighbourhood_basis" class="mw-redirect" title="Neighbourhood basis">neighbourhood basis</a> consisting of compact open subgroups, then it is locally compact and totally disconnected.<sup id="cite_ref-BushnellHenniart_2-1" class="reference"><a href="#cite_note-BushnellHenniart-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Tidy_subgroups">Tidy subgroups</h3></div>

<p>Let <i>G</i> be a locally compact, totally disconnected group, <i>U</i> a compact open subgroup of <i>G</i> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> a continuous automorphism of <i>G</i>.

</p><p>Define:

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<annotation encoding="application/x-tex">{\displaystyle U_{+}=\bigcap _{n\geq 0}\alpha ^{n}(U)}</annotation>

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</math></span><img src="./29df1afbc7c084e58e985c30328dd4a7ce539580.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.969ex; height:5.676ex;" alt="{\displaystyle U_{+}=\bigcap _{n\geq 0}\alpha ^{n}(U)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle U_{-}=\bigcap _{n\geq 0}\alpha ^{-n}(U)}</annotation>

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</math></span><img src="./5b2a627ddc4af562a6a90b2c86dcac812cbdd5bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.247ex; height:5.676ex;" alt="{\displaystyle U_{-}=\bigcap _{n\geq 0}\alpha ^{-n}(U)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle U_{++}=\bigcup _{n\geq 0}\alpha ^{n}(U_{+})}</annotation>

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</math></span><img src="./148ac4d68aea53223fb332d44191d559621ebe6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.563ex; height:5.676ex;" alt="{\displaystyle U_{++}=\bigcup _{n\geq 0}\alpha ^{n}(U_{+})}" loading="lazy"></span></dd>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle U_{--}=\bigcup _{n\geq 0}\alpha ^{-n}(U_{-})}</annotation>

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</math></span><img src="./f651998ab9920f0329d5e3454b104f13d0c892b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.841ex; height:5.676ex;" alt="{\displaystyle U_{--}=\bigcup _{n\geq 0}\alpha ^{-n}(U_{-})}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=U_{+}U_{-}=U_{-}U_{+}}">

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=U_{+}U_{-}=U_{-}U_{+}}</annotation>

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</math></span><img src="./24f4dd21679f9897cead72ea23f31ce85bb79ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.373ex; height:2.509ex;" alt="{\displaystyle U=U_{+}U_{-}=U_{-}U_{+}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{++}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{++}}</annotation>

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</math></span><img src="./ce539b273c08f1fea3cc7cc47ca55aec2749f7f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.377ex; height:2.509ex;" alt="{\displaystyle U_{++}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{--}}">

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<annotation encoding="application/x-tex">{\displaystyle U_{--}}</annotation>

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</math></span><img src="./10ae27b06873235bb795c3f58db076adbee827ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.377ex; height:2.509ex;" alt="{\displaystyle U_{--}}" loading="lazy"></span> are closed.

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<div class="mw-heading mw-heading3"><h3 id="The_scale_function">The scale function</h3></div>

<p>The index of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (U_{+})}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha (U_{+})}</annotation>

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</math></span><img src="./551ba73d2e7834f76ce1e540e7e58a40ce00f8dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.395ex; height:2.843ex;" alt="{\displaystyle \alpha (U_{+})}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{+}}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle U_{+}}</annotation>

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</math></span><img src="./a7780c4d20ba7fd853c57744644d2fb022b2ab17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.509ex;" alt="{\displaystyle U_{+}}" loading="lazy"></span> is shown to be finite and independent of the <i>U</i> which is tidy for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>. Define the scale function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(\alpha )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(\alpha )}</annotation>

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</math></span><img src="./c64d83c90f78a01367aa0266e66204997e0fd64e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle s(\alpha )}" loading="lazy"></span> as this index. Restriction to <a href="Inner_automorphism" title="Inner automorphism">inner automorphisms</a> gives a function on <i>G</i> with interesting properties. These are in particular:<br>

Define the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> on <i>G</i> by

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle s(x):=s(\alpha _{x})}</annotation>

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</math></span><img src="./80a2f81a37f4f416b98734caa7abeca75bbdd6a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.535ex; height:2.843ex;" alt="{\displaystyle s(x):=s(\alpha _{x})}" loading="lazy"></span>,

where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{x}}</annotation>

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</math></span><img src="./6a6aea4f482cc815dc367ff2686b84188beb9ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.66ex; height:2.009ex;" alt="{\displaystyle \alpha _{x}}" loading="lazy"></span> is the inner automorphism of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> on <i>G</i>.

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<div class="mw-heading mw-heading4"><h4 id="Properties">Properties</h4></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> is continuous.</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle s(x)=1}</annotation>

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</math></span><img src="./f1a7893cf7ef3b63180738ceea809508cff566e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.49ex; height:2.843ex;" alt="{\displaystyle s(x)=1}" loading="lazy"></span>, whenever x in <i>G</i> is a compact element.</li>

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<annotation encoding="application/x-tex">{\displaystyle s(x^{n})=s(x)^{n}}</annotation>

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</math></span><img src="./5691080d3ee52e64c23c285114c7c95c2a0781e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.994ex; height:2.843ex;" alt="{\displaystyle s(x^{n})=s(x)^{n}}" loading="lazy"></span> for every non-negative integer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>.</li>

<li>The modular function on <i>G</i> is given by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x)=s(x)s(x^{-1})^{-1}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Delta (x)=s(x)s(x^{-1})^{-1}}</annotation>

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</math></span><img src="./e22d6b83888c16dcdbb1d70dadd1e69b73a4795c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.298ex; height:3.176ex;" alt="{\displaystyle \Delta (x)=s(x)s(x^{-1})^{-1}}" loading="lazy"></span>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Calculations_and_applications">Calculations and applications</h3></div>

<p>The scale function was used to prove a conjecture by Hofmann and Mukherja and has been explicitly calculated for <a href="P-adic" class="mw-redirect" title="P-adic">p-adic</a> <a href="Lie_group" title="Lie group">Lie groups</a> and linear groups over local skew fields by Helge Glöckner.

</p>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<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="#CITEREFCartier1979">Cartier 1979</a>, §1.1</span>

</li>

<li id="cite_note-BushnellHenniart-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-BushnellHenniart_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BushnellHenniart_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBushnellHenniart2006">Bushnell & Henniart 2006</a>, §1.1</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFBorelWallach2000">Borel & Wallach 2000</a>, Chapter X</span>

</li>

<li id="cite_note-Dantzig-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dantzig_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvan_Dantzig1936">van Dantzig 1936</a>, p. 411</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="#CITEREFWillis1994">Willis 1994</a></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFCapraceMonod2011">Caprace & Monod 2011</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFvan_Dantzig1936" class="citation cs2"><a href="David_van_Dantzig" title="David van Dantzig">van Dantzig, David</a> (1936), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=CM_1936__3__408_0">"Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen"</a>, <i><a href="Compositio_Mathematica" title="Compositio Mathematica">Compositio Mathematica</a></i>, <b>3</b>: <span class="nowrap">408–</span>426</cite></li>

<li><cite id="CITEREFBorelWallach2000" class="citation cs2"><a href="Armand_Borel" title="Armand Borel">Borel, Armand</a>; <a href="Nolan_Wallach" title="Nolan Wallach">Wallach, Nolan</a> (2000), <i>Continuous cohomology, discrete subgroups, and representations of reductive groups</i>, Mathematical surveys and monographs, vol. 67 (Second ed.), Providence, Rhode Island: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-0851-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1721403">1721403</a></cite></li>

<li><cite id="CITEREFBushnellHenniart2006" class="citation cs2">Bushnell, Colin J.; Henniart, Guy (2006), <i>The local Langlands conjecture for GL(2)</i>, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-31511-X">10.1007/3-540-31511-X</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-31486-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2234120">2234120</a></cite></li>

<li><cite id="CITEREFCapraceMonod2011" class="citation cs2">Caprace, Pierre-Emmanuel; Monod, Nicolas (2011), "Decomposing locally compact groups into simple pieces", <i><a href="Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a></i>, <b>150</b> (1): <span class="nowrap">97–</span>128, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.4101">0811.4101</a></span>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011MPCPS.150...97C">2011MPCPS.150...97C</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004110000368">10.1017/S0305004110000368</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2739075">2739075</a></cite></li>

<li><cite id="CITEREFCartier1979" class="citation cs2 cs1-prop-long-vol"><a href="Pierre_Cartier_(mathematician)" title="Pierre Cartier (mathematician)">Cartier, Pierre</a> (1979), "Representations of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math></span><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"></span>-adic groups: a survey", in <a href="Armand_Borel" title="Armand Borel">Borel, Armand</a>; <a href="William_Casselman_(mathematician)" class="mw-redirect" title="William Casselman (mathematician)">Casselman, William</a> (eds.), <a rel="nofollow" class="external text" href="http://www.ams.org/online_bks/pspum331/pspum331-ptI-7.pdf"><i>Automorphic Forms, Representations, and L-Functions</i></a> <span class="cs1-format">(PDF)</span>, Proceedings of Symposia in Pure Mathematics, vol. 33, Part 1, Providence, Rhode Island: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, pp. <span class="nowrap">111–</span>155, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-1435-2</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0546593">0546593</a></cite></li>

<li><cite id="CITEREFWillis1994" class="citation cs2"><a href="George_A._Willis" title="George A. Willis">Willis, G.</a> (1994), <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002339951">"The structure of totally disconnected, locally compact groups"</a></span>, <i><a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>300</b>: <span class="nowrap">341–</span>363, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01450491">10.1007/BF01450491</a></cite></li></ul></div><div><div class="zim-footer">

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