  We use the theory of cross ratios to construct a real-valued function f of
only three variables with the property that for any finite set A of reals, the
set f(A) = {f(a,b,c):a,b,c \in A} has cardinality at least C|A|^2/log|A|, for
an absolute constant C. Previously-known functions with this property had all
been of four variables.
  We also improve on the state of the art for functions of four variables by
constructing a function g for which g(A) has cardinality at least C|A|^2; the
previously best-achieved bound was C|A|^2/log|A|.
  Finally, we give an example of a five-variable function h for which h(A) has
cardinality at least C|A|^4/log|A|.
  Proving these results depends only on the Szemeredi-Trotter incidence theorem
and an analoguous result for planes due to Edelsbrunner, Guibas and Sharir,
each applied in the Erlangen-type framework of Elekes and Sharir. In particular
the proofs do not employ the Guth-Katz polynomial partitioning technique or the
theory of ruled surfaces.
  Although the growth exponents for f, g and h are stronger than those for
previously considered functions, it is not clear that they are necessarily
sharp. So we pose a question as to whether the bounds on the cardinalities of
f(A), g(A) and h(A) can be further strengthened.
