Let $G$ be a locally compact unimodular group and
$\mu$ an adapted spread out probability measure on $G$. We relate
the rate of decay of the concentration functions associated with $\mu$
to the growth of a certain subgroup $N_{\mu}$ of $G$. In particular,
we show that
when $\mu$ is strictly aperiodic (i.e., when $N_{\mu}=G$) and $G$
satisfies the growth condition $V_G(m)\geq Cm^D$, then for any compact
neighborhood $K\subset G$ we have
$\sup_{g\in G}\mu^{*n}(gK)
\leq
C'n^{-D/2}$. This extends recent results of Retzlaff \cite{Retzlaff}
on discrete groups for adapted probability
measures.