Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual convolution, i.e., $(\sum\_{g\in G}\lambda\_gg)(\sum\_{h\in G}\mu\_hh)=\sum\_{g, h\in G}\lambda\_g\mu\_hgh$.


Can we find $l=p\_1(y, z)x^{n\_1}+\cdots+ p\_k(y,z)x^{n\_k}\in l^1(G)^{\times}$ such that $$\sum\_{i=1}^k2^{n\_i}p\_i(y,z)(1-z^{n\_i}y)=0\;?$$


Here, $\forall~ 1\leq i\leq k, ~p\_i(y,z)\in \mathbb{Z}G$ 
and $n\_1<\cdots<n\_k\in\mathbb{Z}$ to be determined. 
Note that the group element $x$ does not appear in $p\_i(y, z)$. 




---


Remarks:


1, This problem was asked previously in MSE, but no answer appeared, so I think it might be suitable for MO.


2, This problem is related to a modified Ore condition. I want to show that $l$ does not exist, suppose it exists, then I have considered the natural quotient map $\phi: G\to H=G/\langle z^2\rangle$. Note that it would induce a map $\phi: l^1(G)^{\times}\to l^1(H)^{\times}$, then $\phi(l)\in l^1(H)^{\times}$, but I still could not handle this.. 



Find a special element in group algebra
Omit this problem, we can take $l=z+\frac{1}{2}(-z-1)x+\frac{1}{4}x^2$, then $l^{-1}=\sum\_{n\geq 0, m\geq 0}\frac{1}{2^{n+m}}x^{n+m}z^{-m-1}\in l^1(G)$.


