Is there a nonunimodular group for which Wiener's Tauberian theorem is true? 


Is a locally compact topological group whose volume grows polynomially with radius always unimodular?



Wiener Tauberian Theorem for nonunimodular group
The answer to the second question is yes. Let us show that a non-unimodular, locally compact group $G$ cannot have polynomial growth. Let $\mu$ be left Haar measure, $\Delta$ be the modular function, so that $\mu(Ag)=\mu(A)\Delta(g)$ for $A$ a Borel subset in $G$. Now take for $A$ a compact neighborhood of identity, and $g\in G$ such that $\Delta(g)>1$. Then $Ag^n\subset (Ag)^n$, so $\mu(A)\Delta(g)^n\leq\mu((Ag)^n)$, hence if $\Omega$ is a compact neighborhood of identity containing $Ag$, we have $\mu(A)\Delta(g)^n\leq\mu(\Omega^n)$, and the sequence $(\mu(\Omega^n))\_{n\geq 1}$ has exponential growth.


