Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the
left-invariant Riemannian symmetric space structure and the right Haar measure
$\rho$, which is a Lie group of exponential growth. Hebisch and Steger in
[Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,\rho)$
admits a Calder\'on--Zygmund decomposition which involves a particular family
of sets, called Calder\'on--Zygmund sets. In this paper, we first show the
existence of a dyadic grid in the group $S$, which has {nice} properties
similar to the classical Euclidean dyadic cubes.

Let $L$ be the divergence form elliptic operator with complex bounded
measurable coefficients, $\omega$ the positive concave function on $(0,\infty)$
of strictly critical lower type $p_\oz\in (0, 1]$ and
$\rho(t)={t^{-1}}/\omega^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper,
the authors study the Orlicz-Hardy space $H_{\omega,L}({\mathbb R}^n)$ and its
dual space $\mathrm{BMO}_{\rho,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes
the adjoint operator of $L$ in $L^2({\mathbb R}^n)$.

Let ${\mathcal X}$ be an RD-space, which means that ${\mathcal X}$ is a space
of homogenous type in the sense of Coifman and Weiss with the additional
property that a reverse doubling property holds in ${\mathcal X}$.