Generators of the mapping class group for surfaces with punctures and boundariesSubgroup of mapping class group generated by two Dehn twistsmapping class group of a surfaceMapping class group of a punctured genus 0 surfaceSubgroups of the mapping class group of a surface generated by Dehn twistsExplicit description (=pictures!) of elements in $Mod_g[k]$?Reference request: Mapping class group action on homology of surface with boundarygenerators for the handlebody group of genus twoDehn-Nielsen-Baer Theorem for surfaces with boundary and puncturesIs every element of $Mod(S_g,1)$ a composition of right handed Dehn twists?Mapping Class Group and Triangulations
Multi tool use
Generators of the mapping class group for surfaces with punctures and boundaries
Subgroup of mapping class group generated by two Dehn twistsmapping class group of a surfaceMapping class group of a punctured genus 0 surfaceSubgroups of the mapping class group of a surface generated by Dehn twistsExplicit description (=pictures!) of elements in $Mod_g[k]$?Reference request: Mapping class group action on homology of surface with boundarygenerators for the handlebody group of genus twoDehn-Nielsen-Baer Theorem for surfaces with boundary and puncturesIs every element of $Mod(S_g,1)$ a composition of right handed Dehn twists?Mapping Class Group and Triangulations
$begingroup$
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:
- 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.
- 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
asked yesterday
FKranholdFKranhold
1746
$endgroup$
add a comment |
$begingroup$
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:
- 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.
- 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
asked yesterday
FKranholdFKranhold
1746
$endgroup$
add a comment |
$begingroup$
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:
- 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.
- 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
asked yesterday
FKranholdFKranhold
1746
$endgroup$
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:
- 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.
- 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
asked yesterday
FKranholdFKranhold
1746
asked yesterday
FKranholdFKranhold
1746
edited yesterday
FKranhold
asked yesterday
FKranholdFKranhold
1746
asked yesterday
FKranholdFKranhold
1746
asked yesterday
FKranholdFKranhold
1746
1746
add a comment |
add a comment |
1 Answer
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$begingroup$
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.
answered yesterday
Autumn KentAutumn Kent
9,59734574
$endgroup$
$begingroup$
Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_g, b+mto PGamma^m_g, b$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_g, b^m$ itself?
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
Okay, so these are $binomm2$ additional generators, right? (maybe not all necessary)
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
add a comment |
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$begingroup$
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.
answered yesterday
Autumn KentAutumn Kent
9,59734574
$endgroup$
$begingroup$
Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_g, b+mto PGamma^m_g, b$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_g, b^m$ itself?
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
Okay, so these are $binomm2$ additional generators, right? (maybe not all necessary)
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
add a comment |
$begingroup$
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.
answered yesterday
Autumn KentAutumn Kent
9,59734574
$endgroup$
$begingroup$
Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_g, b+mto PGamma^m_g, b$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_g, b^m$ itself?
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
Okay, so these are $binomm2$ additional generators, right? (maybe not all necessary)
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
add a comment |
$begingroup$
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.
answered yesterday
Autumn KentAutumn Kent
9,59734574
$endgroup$
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.
answered yesterday
Autumn KentAutumn Kent
9,59734574
answered yesterday
Autumn KentAutumn Kent
9,59734574
answered yesterday
Autumn KentAutumn Kent
9,59734574
answered yesterday
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 $Gamma_g, b+mto PGamma^m_g, b$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_g, b^m$ itself?
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
Okay, so these are $binomm2$ additional generators, right? (maybe not all necessary)
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
add a comment |
$begingroup$
Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_g, b+mto PGamma^m_g, b$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_g, b^m$ itself?
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
Okay, so these are $binomm2$ additional generators, right? (maybe not all necessary)
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_g, b+mto PGamma^m_g, b$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_g, b^m$ itself?
$endgroup$
– FKranhold
yesterday
$begingroup$
Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $Gamma_g, b+mto PGamma^m_g, b$ onto the pure MCG, so we can also describe generators there, but what about $Gamma_g, b^m$ itself?
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
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
yesterday
$begingroup$
Okay, so these are $binomm2$ additional generators, right? (maybe not all necessary)
$endgroup$
– FKranhold
yesterday
$begingroup$
Okay, so these are $binomm2$ additional generators, right? (maybe not all necessary)
$endgroup$
– FKranhold
yesterday
$begingroup$
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
yesterday
$begingroup$
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
yesterday
add a comment |
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