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I am wondering if

Let $BS(1,4) = \langle a,b\mid bab^{-1}=a^4 \rangle$ can G$be embedded into the following a group generated by$a_0, a_1, a_2$with relations:$a_0 a_1 a_0^{-1}=a_1^4a_1 a_2 a_1^{-1}=a_2^4a_2 a_0 a_2^{-1}=a_0^4$I am wondering if$BS(1,4)=\langle a,b:bab^{-1}=a^4\rangle$is embedded into G via$a\mapsto a_1$,$b\mapsto a_0$Remark: the group is constructed in analogy to Higman group 4 added 5 characters in body Let I am wondering if$G$BS(1,4) = \langle a,b\mid bab^{-1}=a^4 \rangle$ can be a embedded into the following group generated by $a_0, a_1, a_2$ with relations:

$a_0 a_1 a_0^{-1}=a_1^4$

$a_1 a_2 a_1^{-1}=a_2^4$

$a_2 a_0 a_2^{-1}=a_0^4$

I am wondering if $BS(1,4)=$ is embedded into G via $a\mapsto a_1$, $b\mapsto a_0$

Remark: the group is constructed in analogy to Higman group

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I am wondering if

Let $BS(1,4)= \langle a,b:bab^{-1}=a^4 \rangle$ can G$be embedded into the following a group generated by$a_0, a_1, a_2$with relations:$a_0 a_1 a_0^{-1}=a_1^4a_1 a_2 a_1^{-1}=a_2^4a_2 a_0 a_2^{-1}=a_0^4$I am wondering if$BS(1,4)=$is embedded into G via$a\mapsto a_1$,$b\mapsto a_0\$

Remark: the group is constructed in analogy to Higman group

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