  We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let
$T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes
determined by $(k+1)$-tuples of points from $E$. We prove that there exists
$s_0(d)<d$ such that, if $E \subset {\Bbb R}^d,\, d \ge 2$, with
$dim_{{\mathcal H}}(E)>s_0(d)$, then the ${k+1 \choose 2}$-dimensional Lebesgue
measure of $T_k(E)$ is positive. Results were previously obtained for triangles
in the plane \cite{GI12} and in higher dimensions \cite{GGIP12}. In this paper,
we improve upon those exponents, using a group-theoretic method that sheds new
light on the classical approach to these problems. The key to our approach is a
group action perspective which leads to natural and effective formulae related
to the classical Mattila integral.
