<div dir="ltr"><p dir="ltr"><span style="font-size:12.8px">BINGHAMTON GEOMETRY/TOPOLOGY SEMINAR </span>(joint with Algebra seminar)</p><p dir="ltr" style="font-size:12.8px">   Date: Tuesday, October 13th, 2015</p><p dir="ltr" style="font-size:12.8px">   Time:  2.50 pm</p><p dir="ltr" style="font-size:12.8px">   Place: Whitney Hall, Room 100E,  followed by coffee/tea in the<br>      Hilton Reading Room. Note the new location of the math department (<a href="http://www2.math.binghamton.edu/p/directions" target="_blank">http://www2.</a><a href="http://www2.math.binghamton.edu/p/directions" target="_blank">math.binghamton.edu</a><a href="http://www2.math.binghamton.edu/p/directions" target="_blank">/p/directions</a>).</p><p dir="ltr" style="font-size:12.8px">   Speaker:  <strong style="color:rgb(51,51,51);font-family:Arial,sans-serif;line-height:16.8px">Robert Bieri</strong><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;line-height:16.8px"> (Binghamton University) </span> </p><p dir="ltr" style="font-size:12.8px"><font color="#333333" style="font-size:12.8px">   Title: </font><span style="font-size:12.8px"> </span><strong style="color:rgb(51,51,51);font-family:Arial,sans-serif;line-height:16.8px">Groups of piecewise isometric permutations of lattice points</strong></p><p dir="ltr" style="font-size:12.8px"><span style="font-size:12.8px">    Abstract: </span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;line-height:16.8px">An orthant (of the orthogonal integral lattice)</span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;line-height:16.8px"> </span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:4.003em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.336em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/004C.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/2286.png" class="" style="border: 0px; vertical-align: -3px; color: rgb(102, 102, 102); font-style: italic; min-height: 15px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 13px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.225em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/AMS/Regular/400/005A.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/006E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 6px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.263em;overflow:hidden;width:0px;min-height:1.203em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> is the image of the standard orthant </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.503em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.225em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.225em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/AMS/Regular/400/004E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 13px; display: inline !important; float: none !important; background-color: transparent;"></span></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/006E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 6px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.063em;overflow:hidden;width:0px;min-height:1.07em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> under an affine-orthogonal transformation. And a permutation </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:6.003em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:5.003em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0070.png" class="" style="border: 0px; vertical-align: -4px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; max-height: none; width: 10px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/003A.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 8px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 4px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/27F6.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 10px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 29px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.33em;overflow:hidden;width:0px;min-height:1.27em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px">  of a subset </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.947em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.281em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/2286.png" class="" style="border: 0px; vertical-align: -3px; color: rgb(102, 102, 102); font-style: italic; min-height: 15px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 13px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.225em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/AMS/Regular/400/005A.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/006E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 6px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.263em;overflow:hidden;width:0px;min-height:1.203em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px">  is piecewise-Euclidean-isometric (pei), if </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.836em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.669em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.063em;overflow:hidden;width:0px;min-height:1.07em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> is a disjoint union of finitely many orthants, </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:5.281em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:4.392em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png" class="" style="border: 0px; vertical-align: 2px; color: rgb(102, 102, 102); font-style: italic; min-height: 4px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 13px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.169em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0.003em"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Size1/Regular/400/22C3.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 14px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 4px; display: inline !important; float: none !important; background-color: transparent;"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.169em;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.058em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/004C.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 4px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.463em;overflow:hidden;width:0px;min-height:1.403em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px">, on each of which </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.614em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.503em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0070.png" class="" style="border: 0px; vertical-align: -4px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; max-height: none; width: 10px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.33em;overflow:hidden;width:0px;min-height:0.937em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> restricts to an isometric embedding </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:4.003em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.336em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.058em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/004C.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 4px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/2192.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 10px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 18px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.281em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.263em;overflow:hidden;width:0px;min-height:1.203em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px">. I will talk about the group </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.503em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:2.892em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0070.png" class="" style="border: 0px; vertical-align: -4px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 10px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0065.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0069.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 5px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.397em;overflow:hidden;width:0px;min-height:1.403em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px">  of all pei-permutations of various subsets </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.836em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.669em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.063em;overflow:hidden;width:0px;min-height:1.07em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> and its subgroup </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.614em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.003em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0070.png" class="" style="border: 0px; vertical-align: -4px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 10px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0065.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0074.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.397em;overflow:hidden;width:0px;min-height:1.403em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> of all piecewise-Eucliden-translation permutations. In the case when </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.836em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.669em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.063em;overflow:hidden;width:0px;min-height:1.07em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> consists of the lattice points on the union of </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.781em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.614em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/006E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 11px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.063em;overflow:hidden;width:0px;min-height:0.67em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> positive coordinate axes then </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.614em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.003em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0070.png" class="" style="border: 0px; vertical-align: -4px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 10px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0065.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0074.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.397em;overflow:hidden;width:0px;min-height:1.403em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px"> is the Houghton group </span><span style="font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px;color:rgb(136,136,136)"></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.669em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.392em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.392em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0048.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; max-height: none; width: 17px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.058em"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/006E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 6px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.263em;overflow:hidden;width:0px;min-height:1.203em"></span></span></span><span style="color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:12.8px;line-height:16.8px">.</span></p><p style="font-size:12.8px;line-height:16.8px;color:rgb(51,51,51);font-family:Arial,sans-serif">This is joint work with Heike Sach: we prove finiteness properties of some pei- and pet- groups, and have, as a consequence, that <span style="color:rgb(136,136,136)"></span><span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:4.169em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.447em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0070.png" class="" style="border: 0px; vertical-align: -4px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 10px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0065.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0069.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 5px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.225em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/AMS/Regular/400/005A.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/006E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 6px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.397em;overflow:hidden;width:0px;min-height:1.403em"></span></span></span> admits a K(G,1)-complex with finite <span style="color:rgb(136,136,136)"></span><span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.614em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.003em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.003em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/283/006E.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 6px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/2013.png" class="" style="border: 0px; vertical-align: 5px; color: rgb(102, 102, 102); font-style: italic; min-height: 1px; max-width: none; padding: 0px; margin: 0px 0.003em 0px 0px; max-height: none; width: 9px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px 0px 0px 0.169em;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0031.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 12px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 8px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.397em;overflow:hidden;width:0px;min-height:1.403em"></span></span></span>-skeleton.</p><p style="font-size:12.8px;line-height:16.8px;color:rgb(51,51,51);font-family:Arial,sans-serif">An interesting point is that prominent groups like Richard Thompson's group <span style="color:rgb(136,136,136)"></span><span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.947em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0.781em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Math/Italic/400/0056.png" class="" style="border: 0px; vertical-align: -1px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; max-height: none; width: 14px; display: inline !important; float: none !important; background-color: transparent;"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;overflow:hidden;min-height:1px;width:0.169em"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.063em;overflow:hidden;width:0px;min-height:1.003em"></span></span></span> crop up when we extend the consideration to the <span style="color:rgb(136,136,136)"></span><span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;border:0px;padding:0px;margin:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.892em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:3.225em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:1.614em;min-height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0053.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 14px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 9px; display: inline !important; float: none !important; background-color: transparent;"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/004C.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 11px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/283/0032.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 9px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/AMS/Regular/400/005A.png" class="" style="border: 0px; vertical-align: 0px; color: rgb(102, 102, 102); font-style: italic; min-height: 13px; max-width: none; padding: 0px; margin: 0px 0.058em 0px 0px; max-height: none; width: 12px; display: inline !important; float: none !important; background-color: transparent;"></span></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px"><img src="https://www2.math.binghamton.edu/js/mathjax-MathJax-727332c/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png" class="" style="border: 0px; vertical-align: -5px; color: rgb(102, 102, 102); font-style: italic; min-height: 19px; max-width: none; padding: 0px; margin: 0px 0.114em 0px 0px; max-height: none; width: 6px; display: inline !important; float: none !important; background-color: transparent;"></span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;width:0px;min-height:2.169em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0.003em;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.397em;overflow:hidden;width:0px;min-height:1.403em"></span></span></span>-lattice in the hyperbolic plane.</p><p style="font-size:12.8px;line-height:16.8px;color:rgb(51,51,51);font-family:Arial,sans-serif"><br></p><p style="line-height:16.8px;color:rgb(51,51,51);font-family:Arial,sans-serif;font-size:15px"><span style="font-size:12.8px;font-family:arial,sans-serif;line-height:normal;color:rgb(34,34,34)">NOTE:  The seminar has a webpage where the semester's program is listed:</span><br></p><p dir="ltr" style="font-size:12.8px"><a href="http://www2.math.binghamton.edu/p/seminars/topsem" target="_blank">http://www2.math.binghamton.edu/p/seminars/topsem</a></p></div>