Free groups: Finding words vanishing in two different situations
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Let $varphi_i:Gto H_i$ for $i=1,2$ be two group homomorphisms. I want to find elements in $mathrm{Kern}(varphi_1)cap mathrm{Kern}(varphi_2)$ which are not contained in the commutator $[G,G]$. Is there any systematic way to do this?
In my special case, I am in the following situation: Let $G=langle a_1,dotsc,a_nrangle$ be a free group and let $H=langle{a_1,dotsc,a_{r}}rangle$ with $r<n$. Now consider words $v_{r+1},dotsc,v_nin H$ and $w_{r+1},dotsc,w_nin H$ and the corresponding two homomorphisms $pi_1,pi_2:Gto H$ sending $a_i$ to $a_i$ for $ile r$ and $a_imapsto v_i$ resp. $a_imapsto w_i$ for $ige r+1$. Can I find an $[G,G]notni gin mathrm{Ker}(pi_1)cap mathrm{Ker}(pi_2)$?
group-theory free-groups
asked Nov 17 at 12:34
FKranhold
1787
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up vote
1
down vote
favorite
Let $varphi_i:Gto H_i$ for $i=1,2$ be two group homomorphisms. I want to find elements in $mathrm{Kern}(varphi_1)cap mathrm{Kern}(varphi_2)$ which are not contained in the commutator $[G,G]$. Is there any systematic way to do this?
In my special case, I am in the following situation: Let $G=langle a_1,dotsc,a_nrangle$ be a free group and let $H=langle{a_1,dotsc,a_{r}}rangle$ with $r<n$. Now consider words $v_{r+1},dotsc,v_nin H$ and $w_{r+1},dotsc,w_nin H$ and the corresponding two homomorphisms $pi_1,pi_2:Gto H$ sending $a_i$ to $a_i$ for $ile r$ and $a_imapsto v_i$ resp. $a_imapsto w_i$ for $ige r+1$. Can I find an $[G,G]notni gin mathrm{Ker}(pi_1)cap mathrm{Ker}(pi_2)$?
group-theory free-groups
asked Nov 17 at 12:34
FKranhold
1787
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $varphi_i:Gto H_i$ for $i=1,2$ be two group homomorphisms. I want to find elements in $mathrm{Kern}(varphi_1)cap mathrm{Kern}(varphi_2)$ which are not contained in the commutator $[G,G]$. Is there any systematic way to do this?
In my special case, I am in the following situation: Let $G=langle a_1,dotsc,a_nrangle$ be a free group and let $H=langle{a_1,dotsc,a_{r}}rangle$ with $r<n$. Now consider words $v_{r+1},dotsc,v_nin H$ and $w_{r+1},dotsc,w_nin H$ and the corresponding two homomorphisms $pi_1,pi_2:Gto H$ sending $a_i$ to $a_i$ for $ile r$ and $a_imapsto v_i$ resp. $a_imapsto w_i$ for $ige r+1$. Can I find an $[G,G]notni gin mathrm{Ker}(pi_1)cap mathrm{Ker}(pi_2)$?
group-theory free-groups
asked Nov 17 at 12:34
FKranhold
1787
Let $varphi_i:Gto H_i$ for $i=1,2$ be two group homomorphisms. I want to find elements in $mathrm{Kern}(varphi_1)cap mathrm{Kern}(varphi_2)$ which are not contained in the commutator $[G,G]$. Is there any systematic way to do this?
In my special case, I am in the following situation: Let $G=langle a_1,dotsc,a_nrangle$ be a free group and let $H=langle{a_1,dotsc,a_{r}}rangle$ with $r<n$. Now consider words $v_{r+1},dotsc,v_nin H$ and $w_{r+1},dotsc,w_nin H$ and the corresponding two homomorphisms $pi_1,pi_2:Gto H$ sending $a_i$ to $a_i$ for $ile r$ and $a_imapsto v_i$ resp. $a_imapsto w_i$ for $ige r+1$. Can I find an $[G,G]notni gin mathrm{Ker}(pi_1)cap mathrm{Ker}(pi_2)$?
group-theory free-groups
group-theory free-groups
asked Nov 17 at 12:34
FKranhold
1787
asked Nov 17 at 12:34
FKranhold
1787
asked Nov 17 at 12:34
FKranhold
1787
asked Nov 17 at 12:34
FKranhold
1787
asked Nov 17 at 12:34
FKranhold
1787
1787
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