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?
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? 


The answer to the second question is yes. Let us show that a nonunimodular, 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. 


To the first is yes also. The example is already given above: ax+b group. It is the semidirect product $\mathbb R^\times \ltimes \mathbb R$. Leptin has proved in Leptin, H., Ideal theory in group algebras of locally compact groups. Invent.Math. 31 (1975/76), no. 3, 259278 that every semidirect product of separable abelian l.c. groups has the Wiener property (in the sense that every proper closed twosided ideal is annihilated by a nonzero *representation). 

