On Jones' subgroup of R. Thompson group F

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• 作者：Gili Golan¹ ; ^{gili.golan@math.biu.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Mark Sapir ; ² ; ^{m.sapir@vanderbilt.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：R. Thompson group ; Diagram groups ; Tree-diagrams ; Knots and links
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：122-159
• 全文大小：1853 K

文摘

Recently Vaughan Jones showed that the R. Thompson group F   encodes in a natural way all knots and links in an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si1.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=7d61cdbf8523b68be316608182562497" title="Click to view the MathML source">R³an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="double-struck">R3ath>an>an>an>, and a certain subgroup an id="mmlsi148" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si148.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">ass="imgLazyJSB inlineImage" height="19" width="17" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si148.gif" overflow="scroll">accent="true">F&rarr;ath>an>an>an> of F   encodes all oriented knots and links. We answer several questions of Jones about an id="mmlsi148" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si148.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">ass="imgLazyJSB inlineImage" height="19" width="17" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si148.gif" overflow="scroll">accent="true">F&rarr;ath>an>an>an>. In particular we prove that the subgroup an id="mmlsi148" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si148.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">ass="imgLazyJSB inlineImage" height="19" width="17" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si148.gif" overflow="scroll">accent="true">F&rarr;ath>an>an>an> is generated by an id="mmlsi18" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si18.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=4030a76762529c76cdef2f96cba665e6" title="Click to view the MathML source">x₀x₁an>an class="mathContainer hidden">an class="mathCode">ath altimg="si18.gif" overflow="scroll">x0x1ath>an>an>an>, an id="mmlsi19" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si19.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=4ddd595c51acc61ad3f3bdacb24f5b33" title="Click to view the MathML source">x₁x₂an>an class="mathContainer hidden">an class="mathCode">ath altimg="si19.gif" overflow="scroll">x1x2ath>an>an>an>, an id="mmlsi20" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si20.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=21efd567dc7a00799a40f3f671eec573" title="Click to view the MathML source">x₂x₃an>an class="mathContainer hidden">an class="mathCode">ath altimg="si20.gif" overflow="scroll">x2x3ath>an>an>an> (where an id="mmlsi21" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si21.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=341d6a87b69736dc3469259e9ecfb8d2" title="Click to view the MathML source">x_ian>an class="mathContainer hidden">an class="mathCode">ath altimg="si21.gif" overflow="scroll">xiath>an>an>an>, an id="mmlsi22" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si22.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=4f7aeb42098acd4b4278ed06a0943379" title="Click to view the MathML source">i∈Nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si22.gif" overflow="scroll">i∈athvariant="double-struck">Nath>an>an>an> are the standard generators of F  ) and is isomorphic to an id="mmlsi23" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si23.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=d2033ad416883e85f51b4d146c54f426" title="Click to view the MathML source">F₃an>an class="mathContainer hidden">an class="mathCode">ath altimg="si23.gif" overflow="scroll">F3ath>an>an>an>, the analog of F   where all slopes are powers of 3 and break points are 3-adic rationals. We also show that an id="mmlsi148" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si148.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">ass="imgLazyJSB inlineImage" height="19" width="17" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si148.gif" overflow="scroll">accent="true">F&rarr;ath>an>an>an> coincides with its commensurator. Hence the linearization of the permutational representation of F   on an id="mmlsi9" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si9.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=e2e643d7b1a3688eef794be55cfaead5">ass="imgLazyJSB inlineImage" height="23" width="37" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si9.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">False">/accent="true">F&rarr;ath>an>an>an> is irreducible. We show how to replace 3 in the above results by an arbitrary n, and to construct a series of irreducible representations of F defined in a similar way. Finally we analyze Jones' construction and deduce that the Thompson index of a link is linearly bounded in terms of the number of crossings in a link diagram.