In my work this week I came across a group with presentation with two generators a $a$ and b $b$ subject to the relations baba=1, a^2b=ba^2, $baba=1$, $a^2b=ba^2$, and ab^{-n}ab^n=b^nab^{-n}a. $ab^{-n}ab^n=b^nab^{-n}a$. This group looks like the lamplighter group or something to me, but I couldn't get a sequence of Tietze transformations from this group to the standard presentation for the lamplighter. Does anyone know what this group is? thanks.

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In my work this week I came across a group with presentation twith wo with two generators a and b subject to the relations baba=1, a^2b=ba^2, and ab^{-n}ab^n=b^nab^{-n}a. This group looks like the lamplighter group or something to me, but I couldn't get a sequence of Tietze transformations from this group to the standard presentation for the lamplighter. Does anyone know what this group is? thanks.

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# does this group have a name?

In my work this week I came across a group with presentation twith wo generators a and b subject to the relations baba=1, a^2b=ba^2, and ab^{-n}ab^n=b^nab^{-n}a. This group looks like the lamplighter group or something to me, but I couldn't get a sequence of Tietze transformations from this group to the standard presentation for the lamplighter. Does anyone know what this group is? thanks.