  This paper aims to study in more depth the relation between growth in matrix
groups ${{\rm SL_2}}(\mathbf{F})$ and ${{\rm Aff}}(\mathbf{F})$ over a field
$\mathbf{F}$ by multiplication and geometric incidence estimates, associated
with the sum-product phenomenon over $\mathbf{F}$. It presents streamlined
proofs of Helfgott's theorems on growth in the $\mathbf{F}_p$-case, which avoid
sum-product estimates. For ${{\rm SL_2}}(\mathbf{F}_p)$, for sets exceeding in
size some absolute constant, we improve the lower bound $\frac{1}{1512}$ for
the growth exponent, due to Kowalski, to $\frac{1}{21}.$ For the affine group
we fetch a sharp theorem of Sz\H{o}nyi on the number of directions, determined
by a point set in $\mathbf{F}_p^2$.
  We then focus on ${{\rm Aff}}(\mathbf{F})$ and present a new incidence bound
between a set of points and a set of lines in $\mathbf{F}^2$, which explicitly
depends on the energy of the set of lines as affine transformations under
composition. This bound, strong when the number of lines is considerably
smaller than the number of points, yields generalizations of structural
theorems of Elekes and Murphy on rich lines in grids.
  In the special case when the set of lines is also a grid -- relating back to
sum-products -- we use growth in ${{\rm Aff}}(\mathbf{R})$ to obtain a
subthreshold estimate on the energy of the set of lines. This yields a unified
way to break the ice in various threshold sum-product type energy inequalities.
We show this in applications to energy estimates, corresponding to sets $A(A+
A)$, $A+AA$ (also embracing asymmetric versions) as well as $A+B$ when $A$ has
small multiplicative doubling and $\sqrt{|A|} \le |B|\le|A|^{1+o(1)}$.
