Equivalence of surjections from a surface group to a free group

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Let $g geq 2$. Let $S = langle a_1,b_2,...,a_g,b_g | [a_1,b_1] cdots [a_g,b_g] rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given two surjections $f_1,f_2 : S to F_g$ is there a way to determine if there are automophisms $phi: S to S$ and $psi: F_g to F_g$ so that $f_1 = phi circ f_2 circ psi$?



Is there an example of two surjections $f_1,f_2$ that are not equivalent in the above way?



I asked the question on MSE before but didn't get much.






gr.group-theory gt.geometric-topology free-groups







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  • A naive question: is it clear such a surjection exists?
    – PseudoNeo
    3 hours ago






  • 1




    @PseudoNeo Algebraically, yes: kill all of the $b_i$. Geometrically, yes: the surface is the boundary of a handlebody, equivalent to a wedge of $g$ circles.
    – Mike Miller
    3 hours ago










  • Oh, thank you, I was misreading the question (I mixed up $F_g$ and $F_2g$) and was very confused.
    – PseudoNeo
    3 hours ago














up vote
10
down vote

favorite
3












Let $g geq 2$. Let $S = langle a_1,b_2,...,a_g,b_g | [a_1,b_1] cdots [a_g,b_g] rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given two surjections $f_1,f_2 : S to F_g$ is there a way to determine if there are automophisms $phi: S to S$ and $psi: F_g to F_g$ so that $f_1 = phi circ f_2 circ psi$?



Is there an example of two surjections $f_1,f_2$ that are not equivalent in the above way?



I asked the question on MSE before but didn't get much.






gr.group-theory gt.geometric-topology free-groups







share|cite|improve this question





















  • A naive question: is it clear such a surjection exists?
    – PseudoNeo
    3 hours ago






  • 1




    @PseudoNeo Algebraically, yes: kill all of the $b_i$. Geometrically, yes: the surface is the boundary of a handlebody, equivalent to a wedge of $g$ circles.
    – Mike Miller
    3 hours ago










  • Oh, thank you, I was misreading the question (I mixed up $F_g$ and $F_2g$) and was very confused.
    – PseudoNeo
    3 hours ago












up vote
10
down vote

favorite
3









up vote
10
down vote

favorite
3






3





Let $g geq 2$. Let $S = langle a_1,b_2,...,a_g,b_g | [a_1,b_1] cdots [a_g,b_g] rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given two surjections $f_1,f_2 : S to F_g$ is there a way to determine if there are automophisms $phi: S to S$ and $psi: F_g to F_g$ so that $f_1 = phi circ f_2 circ psi$?



Is there an example of two surjections $f_1,f_2$ that are not equivalent in the above way?



I asked the question on MSE before but didn't get much.






gr.group-theory gt.geometric-topology free-groups







share|cite|improve this question













Let $g geq 2$. Let $S = langle a_1,b_2,...,a_g,b_g | [a_1,b_1] cdots [a_g,b_g] rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given two surjections $f_1,f_2 : S to F_g$ is there a way to determine if there are automophisms $phi: S to S$ and $psi: F_g to F_g$ so that $f_1 = phi circ f_2 circ psi$?



Is there an example of two surjections $f_1,f_2$ that are not equivalent in the above way?



I asked the question on MSE before but didn't get much.






gr.group-theory gt.geometric-topology free-groups




gr.group-theory gt.geometric-topology free-groups






share|cite|improve this question













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asked 5 hours ago









user101010

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  • A naive question: is it clear such a surjection exists?
    – PseudoNeo
    3 hours ago






  • 1




    @PseudoNeo Algebraically, yes: kill all of the $b_i$. Geometrically, yes: the surface is the boundary of a handlebody, equivalent to a wedge of $g$ circles.
    – Mike Miller
    3 hours ago










  • Oh, thank you, I was misreading the question (I mixed up $F_g$ and $F_2g$) and was very confused.
    – PseudoNeo
    3 hours ago
















  • A naive question: is it clear such a surjection exists?
    – PseudoNeo
    3 hours ago






  • 1




    @PseudoNeo Algebraically, yes: kill all of the $b_i$. Geometrically, yes: the surface is the boundary of a handlebody, equivalent to a wedge of $g$ circles.
    – Mike Miller
    3 hours ago










  • Oh, thank you, I was misreading the question (I mixed up $F_g$ and $F_2g$) and was very confused.
    – PseudoNeo
    3 hours ago















A naive question: is it clear such a surjection exists?
– PseudoNeo
3 hours ago




A naive question: is it clear such a surjection exists?
– PseudoNeo
3 hours ago




1




1




@PseudoNeo Algebraically, yes: kill all of the $b_i$. Geometrically, yes: the surface is the boundary of a handlebody, equivalent to a wedge of $g$ circles.
– Mike Miller
3 hours ago




@PseudoNeo Algebraically, yes: kill all of the $b_i$. Geometrically, yes: the surface is the boundary of a handlebody, equivalent to a wedge of $g$ circles.
– Mike Miller
3 hours ago












Oh, thank you, I was misreading the question (I mixed up $F_g$ and $F_2g$) and was very confused.
– PseudoNeo
3 hours ago




Oh, thank you, I was misreading the question (I mixed up $F_g$ and $F_2g$) and was very confused.
– PseudoNeo
3 hours ago










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This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid https://arxiv.org/abs/math/0202261. They state the result in slightly different language, that is they prove that any such epimorphism is induced by choosing a genus $g$ handlebody.






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    This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid https://arxiv.org/abs/math/0202261. They state the result in slightly different language, that is they prove that any such epimorphism is induced by choosing a genus $g$ handlebody.






    share|cite|improve this answer
























      up vote
      3
      down vote













      This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid https://arxiv.org/abs/math/0202261. They state the result in slightly different language, that is they prove that any such epimorphism is induced by choosing a genus $g$ handlebody.






      share|cite|improve this answer






















        up vote
        3
        down vote










        up vote
        3
        down vote









        This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid https://arxiv.org/abs/math/0202261. They state the result in slightly different language, that is they prove that any such epimorphism is induced by choosing a genus $g$ handlebody.






        share|cite|improve this answer












        This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid https://arxiv.org/abs/math/0202261. They state the result in slightly different language, that is they prove that any such epimorphism is induced by choosing a genus $g$ handlebody.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 16 mins ago









        Jean Raimbault

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