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.