Commutators and abelianisations of congruence subgroups in function fields
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Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite field, and let $r>1$ be an integer.
I'm currently looking for the abelianisation of the congruence subgroup $Γ(N)$ of the special linear group $SL(r,A)$, i.e. the kernel of the 'modulo $N$' map $SL(r,A) to SL(r,A/N)$, where $N in A$ is a nonconstant polynomial; more particularly, I'm looking for the torsion-free part of the abelianisation, but I wouldn't complain about knowing the abelianisation itself if that's possible.
So, here are my questions, in reverse order of importance:
- What are the commutator subgroups of $SL(r,A)$ and $Γ(N)$?
- What are the abelianisations of these groups?
- What are the torsion-free abelianisations of these groups?
I wasn't sure whether this should be posted on math.overflow instead; please move it if it should be there instead.
group-theory abelian-groups function-fields
asked Nov 20 at 13:54
Liam Baker
214
add a comment |
up vote
1
down vote
favorite
Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite field, and let $r>1$ be an integer.
I'm currently looking for the abelianisation of the congruence subgroup $Γ(N)$ of the special linear group $SL(r,A)$, i.e. the kernel of the 'modulo $N$' map $SL(r,A) to SL(r,A/N)$, where $N in A$ is a nonconstant polynomial; more particularly, I'm looking for the torsion-free part of the abelianisation, but I wouldn't complain about knowing the abelianisation itself if that's possible.
So, here are my questions, in reverse order of importance:
- What are the commutator subgroups of $SL(r,A)$ and $Γ(N)$?
- What are the abelianisations of these groups?
- What are the torsion-free abelianisations of these groups?
I wasn't sure whether this should be posted on math.overflow instead; please move it if it should be there instead.
group-theory abelian-groups function-fields
asked Nov 20 at 13:54
Liam Baker
214
2
Cross-posted to mathOF: mathoverflow.net/questions/315949/…
– YCor
Nov 22 at 11:58
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite field, and let $r>1$ be an integer.
I'm currently looking for the abelianisation of the congruence subgroup $Γ(N)$ of the special linear group $SL(r,A)$, i.e. the kernel of the 'modulo $N$' map $SL(r,A) to SL(r,A/N)$, where $N in A$ is a nonconstant polynomial; more particularly, I'm looking for the torsion-free part of the abelianisation, but I wouldn't complain about knowing the abelianisation itself if that's possible.
So, here are my questions, in reverse order of importance:
- What are the commutator subgroups of $SL(r,A)$ and $Γ(N)$?
- What are the abelianisations of these groups?
- What are the torsion-free abelianisations of these groups?
I wasn't sure whether this should be posted on math.overflow instead; please move it if it should be there instead.
group-theory abelian-groups function-fields
asked Nov 20 at 13:54
Liam Baker
214
Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite field, and let $r>1$ be an integer.
I'm currently looking for the abelianisation of the congruence subgroup $Γ(N)$ of the special linear group $SL(r,A)$, i.e. the kernel of the 'modulo $N$' map $SL(r,A) to SL(r,A/N)$, where $N in A$ is a nonconstant polynomial; more particularly, I'm looking for the torsion-free part of the abelianisation, but I wouldn't complain about knowing the abelianisation itself if that's possible.
So, here are my questions, in reverse order of importance:
- What are the commutator subgroups of $SL(r,A)$ and $Γ(N)$?
- What are the abelianisations of these groups?
- What are the torsion-free abelianisations of these groups?
I wasn't sure whether this should be posted on math.overflow instead; please move it if it should be there instead.
group-theory abelian-groups function-fields
group-theory abelian-groups function-fields
asked Nov 20 at 13:54
Liam Baker
214
asked Nov 20 at 13:54
Liam Baker
214
asked Nov 20 at 13:54
Liam Baker
214
asked Nov 20 at 13:54
Liam Baker
214
asked Nov 20 at 13:54
Liam Baker
214
214
2
Cross-posted to mathOF: mathoverflow.net/questions/315949/…
– YCor
Nov 22 at 11:58
add a comment |
2
Cross-posted to mathOF: mathoverflow.net/questions/315949/…
– YCor
Nov 22 at 11:58
2
2
Cross-posted to mathOF: mathoverflow.net/questions/315949/…
– YCor
Nov 22 at 11:58
Cross-posted to mathOF: mathoverflow.net/questions/315949/…
– YCor
Nov 22 at 11:58
add a comment |
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Cross-posted to mathOF: mathoverflow.net/questions/315949/…
– YCor
Nov 22 at 11:58