<div>Next week, we will meet twice: Monday at 4:15pm, and Thursday at 2:50 (as usual).</div>
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<div><strong>SPECIAL DATE AND TIME: October 28, 4:15 - 5:15, WH 100E</strong> <br /> Speaker: <strong>Yash Lodha</strong> (EPFL) <br /> Title: <strong>Property FW and smoothability</strong> <em>Abstract: </em>I shall describe joint work with Matte Bon and Triestino. We demonstrate that aperiodic actions of Kazhdan groups by countably singular diffeomorphisms on closed manifolds are smoothable. In the case of the circle, we obtain a proof that groups of piecewise linear or piecewise projective homeomorphisms are not Kazhdan unless they are finite. The key new idea is the application of Property FW, which is a weakening of Kazhdan's property (T).</div>
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<div><strong>October 31, 2:50 - 3:50pm, WH 100E</strong> <br /> Speaker: <strong>Eduard Schesler</strong> (Universität Bielefeld) <br /> Title: <strong>The Sigma conjecture for solvable <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-64" role="math" style="width: 0.617em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.488em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.718em, 1000.49em, 2.728em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-65"><span class="mi" id="MathJax-Span-66" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.089em; border-left: 0px solid; width: 0px; height: 0.976em;"></span></span></nobr></span></strong><strong>-arithmetic groups via discrete Morse theory on Euclidean buildings.</strong>
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<p> <em>Abstract: </em>Given a finitely generated group <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-67" role="math" style="width: 0.883em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.705em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.719em, 1000.71em, 2.728em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-68"><span class="mi" id="MathJax-Span-69" style="font-family: STIXGeneral; font-style: italic;">G</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.089em; border-left: 0px solid; width: 0px; height: 0.975em;"></span></span></nobr></span>, the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Σ</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-70" role="math" style="width: 0.817em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.65em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.723em, 1000.63em, 2.71em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-71"><span class="mi" id="MathJax-Span-72" style="font-family: STIXGeneral;">Σ</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.067em; border-left: 0px solid; width: 0px; height: 0.948em;"></span></span></nobr></span> invariants of <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-73" role="math" style="width: 0.883em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.705em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.719em, 1000.71em, 2.728em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-74"><span class="mi" id="MathJax-Span-75" style="font-family: STIXGeneral; font-style: italic;">G</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.089em; border-left: 0px solid; width: 0px; height: 0.975em;"></span></span></nobr></span> consist of geometrically defined subsets <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">Σ</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-76" role="math" style="width: 2.95em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.385em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.539em, 1002.34em, 2.887em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-77"><span class="msubsup" id="MathJax-Span-78"><span style="display: inline-block; position: relative; width: 1.025em; height: 0px;"><span style="position: absolute; clip: rect(3.186em, 1000.6em, 4.173em, -1000em); top: -4.011em; left: 0em;"><span class="mi" id="MathJax-Span-79" style="font-family: STIXGeneral;">Σ</span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span><span style="position: absolute; top: -4.374em; left: 0.624em;"><span class="mi" id="MathJax-Span-80" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span></span></span><span class="mo" id="MathJax-Span-81" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-82" style="font-family: STIXGeneral; font-style: italic;">G</span><span class="mo" id="MathJax-Span-83" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.284em; border-left: 0px solid; width: 0px; height: 1.391em;"></span></span></nobr></span> of the set <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-84" role="math" style="width: 2.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.843em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.709em, 1001.8em, 2.887em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-85"><span class="mi" id="MathJax-Span-86" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span class="mo" id="MathJax-Span-87" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-88" style="font-family: STIXGeneral; font-style: italic;">G</span><span class="mo" id="MathJax-Span-89" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.284em; border-left: 0px solid; width: 0px; height: 1.183em;"></span></span></nobr></span> of all characters <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi><mo>:</mo><mi>G</mi><mo stretchy="false">→</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">R</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-90" role="math" style="width: 5.35em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.709em, 1004.33em, 2.917em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-91"><span class="mi" id="MathJax-Span-92" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span class="mo" id="MathJax-Span-93" style="font-family: STIXGeneral; padding-left: 0.313em;">:</span><span class="mi" id="MathJax-Span-94" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;">G</span><span class="mo" id="MathJax-Span-95" style="font-family: STIXGeneral; padding-left: 0.313em;">→</span><span class="texatom" id="MathJax-Span-96" style="padding-left: 0.313em;"><span class="mrow" id="MathJax-Span-97"><span class="mi" id="MathJax-Span-98" style="font-family: STIXGeneral; font-weight: bold;">R</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.321em; border-left: 0px solid; width: 0px; height: 1.219em;"></span></span></nobr></span> of <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-99" role="math" style="width: 0.883em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.705em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.719em, 1000.71em, 2.728em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-100"><span class="mi" id="MathJax-Span-101" style="font-family: STIXGeneral; font-style: italic;">G</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.089em; border-left: 0px solid; width: 0px; height: 0.975em;"></span></span></nobr></span>. These invariants were introduced independently by Bieri-Strebel and Neumann for <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>1</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-102" role="math" style="width: 2.817em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.276em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.702em, 1002.17em, 2.721em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-103"><span class="mi" id="MathJax-Span-104" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span class="mo" id="MathJax-Span-105" style="font-family: STIXGeneral; padding-left: 0.313em;">=</span><span class="mn" id="MathJax-Span-106" style="font-family: STIXGeneral; padding-left: 0.313em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.08em; border-left: 0px solid; width: 0px; height: 0.987em;"></span></span></nobr></span> and generalized by Bieri-Renz to the general case in the late 80's in order to determine the finiteness properties of all subgroups <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-107" role="math" style="width: 0.95em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.759em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.732em, 1000.76em, 2.71em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-108"><span class="mi" id="MathJax-Span-109" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.067em; border-left: 0px solid; width: 0px; height: 0.937em;"></span></span></nobr></span> of <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-110" role="math" style="width: 0.883em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.705em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.719em, 1000.71em, 2.728em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-111"><span class="mi" id="MathJax-Span-112" style="font-family: STIXGeneral; font-style: italic;">G</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.089em; border-left: 0px solid; width: 0px; height: 0.975em;"></span></span></nobr></span> that contain the commutator subgroup <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>G</mi><mo stretchy="false">]</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-113" role="math" style="width: 3.083em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.493em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.719em, 1002.41em, 2.866em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-114"><span class="mo" id="MathJax-Span-115" style="font-family: STIXGeneral;">[</span><span class="mi" id="MathJax-Span-116" style="font-family: STIXGeneral; font-style: italic;">G</span><span class="mo" id="MathJax-Span-117" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-118" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;">G</span><span class="mo" id="MathJax-Span-119" style="font-family: STIXGeneral;">]</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.259em; border-left: 0px solid; width: 0px; height: 1.144em;"></span></span></nobr></span>. In this talk we determine the Sigma invariants of certain <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-120" role="math" style="width: 0.617em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.488em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.718em, 1000.49em, 2.728em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-121"><span class="mi" id="MathJax-Span-122" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.089em; border-left: 0px solid; width: 0px; height: 0.976em;"></span></span></nobr></span>-arithmetic subgroups of Borel groups in Chevalley groups. In particular we will determine the finiteness properties of every subgroup of the group of upper triangular matrices <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Z</mi></mrow><mo stretchy="false">[</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>p</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo><</mo><mi>S</mi><msub><mi>L</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Z</mi></mrow><mo stretchy="false">[</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>p</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-123" role="math" style="width: 12.617em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.244em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.709em, 1010.2em, 2.915em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-124"><span class="msubsup" id="MathJax-Span-125"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.195em, 1000.59em, 4.173em, -1000em); top: -4.011em; left: 0em;"><span class="mi" id="MathJax-Span-126" style="font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span><span style="position: absolute; top: -3.861em; left: 0.611em;"><span class="mi" id="MathJax-Span-127" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span></span></span><span class="mo" id="MathJax-Span-128" style="font-family: STIXGeneral;">(</span><span class="texatom" id="MathJax-Span-129"><span class="mrow" id="MathJax-Span-130"><span class="mi" id="MathJax-Span-131" style="font-family: STIXGeneral; font-weight: bold;">Z</span></span></span><span class="mo" id="MathJax-Span-132" style="font-family: STIXGeneral;">[</span><span class="mn" id="MathJax-Span-133" style="font-family: STIXGeneral;">1</span><span class="texatom" id="MathJax-Span-134"><span class="mrow" id="MathJax-Span-135"><span class="mo" id="MathJax-Span-136" style="font-family: STIXGeneral;">/</span></span></span><span class="mi" id="MathJax-Span-137" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-138" style="font-family: STIXGeneral;">]</span><span class="mo" id="MathJax-Span-139" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-140" style="font-family: STIXGeneral; padding-left: 0.313em;"><</span><span class="mi" id="MathJax-Span-141" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span class="msubsup" id="MathJax-Span-142"><span style="display: inline-block; position: relative; width: 0.985em; height: 0px;"><span style="position: absolute; clip: rect(3.195em, 1000.56em, 4.173em, -1000em); top: -4.011em; left: 0em;"><span class="mi" id="MathJax-Span-143" style="font-family: STIXGeneral; font-style: italic;">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span><span style="position: absolute; top: -3.861em; left: 0.556em;"><span class="mi" id="MathJax-Span-144" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span></span></span><span class="mo" id="MathJax-Span-145" style="font-family: STIXGeneral;">(</span><span class="texatom" id="MathJax-Span-146"><span class="mrow" id="MathJax-Span-147"><span class="mi" id="MathJax-Span-148" style="font-family: STIXGeneral; font-weight: bold;">Z</span></span></span><span class="mo" id="MathJax-Span-149" style="font-family: STIXGeneral;">[</span><span class="mn" id="MathJax-Span-150" style="font-family: STIXGeneral;">1</span><span class="texatom" id="MathJax-Span-151"><span class="mrow" id="MathJax-Span-152"><span class="mo" id="MathJax-Span-153" style="font-family: STIXGeneral;">/</span></span></span><span class="mi" id="MathJax-Span-154" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-155" style="font-family: STIXGeneral;">]</span><span class="mo" id="MathJax-Span-156" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.319em; border-left: 0px solid; width: 0px; height: 1.217em;"></span></span></nobr></span> that contains the group <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Z</mi></mrow><mo stretchy="false">[</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>p</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-157" role="math" style="width: 5.35em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.709em, 1004.29em, 2.915em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-158"><span class="msubsup" id="MathJax-Span-159"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.195em, 1000.77em, 4.191em, -1000em); top: -4.011em; left: 0em;"><span class="mi" id="MathJax-Span-160" style="font-family: STIXGeneral; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span><span style="position: absolute; top: -3.861em; left: 0.722em;"><span class="mi" id="MathJax-Span-161" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 4.011em;"></span></span></span></span><span class="mo" id="MathJax-Span-162" style="font-family: STIXGeneral;">(</span><span class="texatom" id="MathJax-Span-163"><span class="mrow" id="MathJax-Span-164"><span class="mi" id="MathJax-Span-165" style="font-family: STIXGeneral; font-weight: bold;">Z</span></span></span><span class="mo" id="MathJax-Span-166" style="font-family: STIXGeneral;">[</span><span class="mn" id="MathJax-Span-167" style="font-family: STIXGeneral;">1</span><span class="texatom" id="MathJax-Span-168"><span class="mrow" id="MathJax-Span-169"><span class="mo" id="MathJax-Span-170" style="font-family: STIXGeneral;">/</span></span></span><span class="mi" id="MathJax-Span-171" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-172" style="font-family: STIXGeneral;">]</span><span class="mo" id="MathJax-Span-173" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.319em; border-left: 0px solid; width: 0px; height: 1.217em;"></span></span></nobr></span> of unipotent matrices where <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-174" role="math" style="width: 0.617em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.488em; height: 0px; font-size: 123%;"><span style="position: absolute; clip: rect(1.944em, 1000.46em, 2.915em, -1000em); top: -2.547em; left: 0em;"><span class="mrow" id="MathJax-Span-175"><span class="mi" id="MathJax-Span-176" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.547em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.319em; border-left: 0px solid; width: 0px; height: 0.928em;"></span></span></nobr></span> is any sufficiently large prime number.</p>
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