Generators of the mapping class group for surfaces with punctures and boundaries












4












$begingroup$


Let $Gamma_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.



It is clear that in general, presentations for these groups are hard. I only need for the general case a set of generators (I want to show that two morphisms $varphi,psi:Gammato G$ are equal and I want to check this on the generators). Partial answers are the following:




  1. If $b,m=0$ (so we have a closed surface), then the group is generated by Dehn twists which can be easily drawn on the surface.

  2. If $g,m=0$, we can declare one boundary curve as “outer” and the group is generated by Dehn twists along the inner boundary curves and the pure braid generators $alpha_{ij}$.


From Farb–Margalit, I know that there are always finitely many Dehn twists (or half twists) which generate $Gamma$, but can we say in general where they are?






moduli-spaces mapping-class-groups surfaces







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$endgroup$

















    4












    $begingroup$


    Let $Gamma_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.



    It is clear that in general, presentations for these groups are hard. I only need for the general case a set of generators (I want to show that two morphisms $varphi,psi:Gammato G$ are equal and I want to check this on the generators). Partial answers are the following:




    1. If $b,m=0$ (so we have a closed surface), then the group is generated by Dehn twists which can be easily drawn on the surface.

    2. If $g,m=0$, we can declare one boundary curve as “outer” and the group is generated by Dehn twists along the inner boundary curves and the pure braid generators $alpha_{ij}$.


    From Farb–Margalit, I know that there are always finitely many Dehn twists (or half twists) which generate $Gamma$, but can we say in general where they are?






    moduli-spaces mapping-class-groups surfaces







    share|cite|improve this question











    $endgroup$















      4












      4








      4


      1



      $begingroup$


      Let $Gamma_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.



      It is clear that in general, presentations for these groups are hard. I only need for the general case a set of generators (I want to show that two morphisms $varphi,psi:Gammato G$ are equal and I want to check this on the generators). Partial answers are the following:




      1. If $b,m=0$ (so we have a closed surface), then the group is generated by Dehn twists which can be easily drawn on the surface.

      2. If $g,m=0$, we can declare one boundary curve as “outer” and the group is generated by Dehn twists along the inner boundary curves and the pure braid generators $alpha_{ij}$.


      From Farb–Margalit, I know that there are always finitely many Dehn twists (or half twists) which generate $Gamma$, but can we say in general where they are?






      moduli-spaces mapping-class-groups surfaces







      share|cite|improve this question











      $endgroup$




      Let $Gamma_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.



      It is clear that in general, presentations for these groups are hard. I only need for the general case a set of generators (I want to show that two morphisms $varphi,psi:Gammato G$ are equal and I want to check this on the generators). Partial answers are the following:




      1. If $b,m=0$ (so we have a closed surface), then the group is generated by Dehn twists which can be easily drawn on the surface.

      2. If $g,m=0$, we can declare one boundary curve as “outer” and the group is generated by Dehn twists along the inner boundary curves and the pure braid generators $alpha_{ij}$.


      From Farb–Margalit, I know that there are always finitely many Dehn twists (or half twists) which generate $Gamma$, but can we say in general where they are?






      moduli-spaces mapping-class-groups surfaces




      moduli-spaces mapping-class-groups surfaces






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago







      FKranhold

















      asked 2 days ago









      FKranholdFKranhold

      1846




      1846






















          1 Answer
          1






          active

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          4












          $begingroup$

          See the paper



          B. Wajnryb, "An elementary approach to the mapping class group of a surface,"
          Geometry & Topology 3 (1999) 405–466.



          See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: https://arxiv.org/pdf/math/9811162.pdf.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_{g, b+m}to PGamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_{g, b}^m$ itself?
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks.
            $endgroup$
            – Autumn Kent
            2 days ago










          • $begingroup$
            Okay, so these are $binom{m}{2}$ additional generators, right? (maybe not all necessary)
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators.
            $endgroup$
            – Autumn Kent
            2 days ago











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          1 Answer
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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          4












          $begingroup$

          See the paper



          B. Wajnryb, "An elementary approach to the mapping class group of a surface,"
          Geometry & Topology 3 (1999) 405–466.



          See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: https://arxiv.org/pdf/math/9811162.pdf.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_{g, b+m}to PGamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_{g, b}^m$ itself?
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks.
            $endgroup$
            – Autumn Kent
            2 days ago










          • $begingroup$
            Okay, so these are $binom{m}{2}$ additional generators, right? (maybe not all necessary)
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators.
            $endgroup$
            – Autumn Kent
            2 days ago
















          4












          $begingroup$

          See the paper



          B. Wajnryb, "An elementary approach to the mapping class group of a surface,"
          Geometry & Topology 3 (1999) 405–466.



          See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: https://arxiv.org/pdf/math/9811162.pdf.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_{g, b+m}to PGamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_{g, b}^m$ itself?
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks.
            $endgroup$
            – Autumn Kent
            2 days ago










          • $begingroup$
            Okay, so these are $binom{m}{2}$ additional generators, right? (maybe not all necessary)
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators.
            $endgroup$
            – Autumn Kent
            2 days ago














          4












          4








          4





          $begingroup$

          See the paper



          B. Wajnryb, "An elementary approach to the mapping class group of a surface,"
          Geometry & Topology 3 (1999) 405–466.



          See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: https://arxiv.org/pdf/math/9811162.pdf.






          share|cite|improve this answer









          $endgroup$



          See the paper



          B. Wajnryb, "An elementary approach to the mapping class group of a surface,"
          Geometry & Topology 3 (1999) 405–466.



          See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: https://arxiv.org/pdf/math/9811162.pdf.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 days ago









          Autumn KentAutumn Kent

          9,59734574




          9,59734574












          • $begingroup$
            Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_{g, b+m}to PGamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_{g, b}^m$ itself?
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks.
            $endgroup$
            – Autumn Kent
            2 days ago










          • $begingroup$
            Okay, so these are $binom{m}{2}$ additional generators, right? (maybe not all necessary)
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators.
            $endgroup$
            – Autumn Kent
            2 days ago


















          • $begingroup$
            Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_{g, b+m}to PGamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_{g, b}^m$ itself?
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks.
            $endgroup$
            – Autumn Kent
            2 days ago










          • $begingroup$
            Okay, so these are $binom{m}{2}$ additional generators, right? (maybe not all necessary)
            $endgroup$
            – FKranhold
            2 days ago










          • $begingroup$
            You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators.
            $endgroup$
            – Autumn Kent
            2 days ago
















          $begingroup$
          Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_{g, b+m}to PGamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_{g, b}^m$ itself?
          $endgroup$
          – FKranhold
          2 days ago




          $begingroup$
          Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_{g, b+m}to PGamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_{g, b}^m$ itself?
          $endgroup$
          – FKranhold
          2 days ago












          $begingroup$
          Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks.
          $endgroup$
          – Autumn Kent
          2 days ago




          $begingroup$
          Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks.
          $endgroup$
          – Autumn Kent
          2 days ago












          $begingroup$
          Okay, so these are $binom{m}{2}$ additional generators, right? (maybe not all necessary)
          $endgroup$
          – FKranhold
          2 days ago




          $begingroup$
          Okay, so these are $binom{m}{2}$ additional generators, right? (maybe not all necessary)
          $endgroup$
          – FKranhold
          2 days ago












          $begingroup$
          You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators.
          $endgroup$
          – Autumn Kent
          2 days ago




          $begingroup$
          You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators.
          $endgroup$
          – Autumn Kent
          2 days ago


















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