  Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be
a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the
"algebraic" local moduli in characteristic p of $G_0$, by $G$ the universal
deformation of $G_0$ over $S$, and by $U\subset S$ the ordinary locus of $G$.
The etale part of $G$ over $U$ gives rise to a monodromy representation $\rho$
of the fundamental group of $U$ on the Tate module of $G$. Motivated by a
famous theorem of Igusa, we prove in this article that $\rho$ is surjective if
$G_0$ is connected and HW-cyclic. This latter condition is equivalent to that
Oort's $a$-number of $G_0$ equals 1, and it is satisfied by all connected
one-dimensional Barsotti-Tate groups over $k$.
