  This paper develops further the theory of the automorphic group of
non-constant entire functions. This theory has already a long history that
essentially started with two remarkable papers of Tatsujir\^o Shimizu that were
published in 1931. The elements $\phi(z)$ of the group are defined by the
automorphic equation $f(\phi(z))=f(z)$, were $f(z)$ is entire. Tatsujir\^o
Shimizu also refers to the functions of this group as those functions that are
determined by $f^{-1}\circ f$. He proved many remarkable properties of those
automorphic functions. He indicated how they induce a beautiful geometric
structure on the complex plane. Those structures were termed by Tatsujir\^o
Shimizu, the system of normal polygonal domains, and the more refined system of
the fundamental domains of $f(z)$. The last system if exists tiles up the
complex plane with remarkable geometric tiles that are conformally mapped to
one another by the automorphic functions. In the Ph.D thesis of the author,
those tiles were also called the system of the maximal domains of $f(z)$. One
can not avoid noticing the many similarities between this automorphic group and
its accompanying geometric structures and analytic properties, and the more
tame discrete groups that appear in the theory of hyperbolic geometry and also
the arithmetic groups in number theory. This paper pursues further the theory
initiated by Tatsujir\^o Shimizu, towards understanding global properties of
the automorphic group, rather than just understanding the properties of the
individual automorphic functions. We hope to be able in sequel papers to
generalize arithmetic and analytic tools such as the Selberg trace formula, to
this new setting.
