  Archdeacon, in his seminal paper $[1]$, defined the concept of Heffter array
in order to provide explicit constructions of $\mathbb{Z}_{v}$-regular
biembeddings of complete graphs $K_v$ into orientable surfaces.
  In this paper, we first introduce the quasi-Heffter arrays as a
generalization of the concept of Heffer array and we show that, in this
context, we can define a $2$-colorable embedding of Archdeacon type of the
complete multipartite graph $K_{\frac{v}{t}\times t}$ into an orientable
surface. Then, our main goal is to study the full automorphism groups of these
embeddings: here we are able to prove, using a probabilistic approach, that,
almost always, this group is exactly $\mathbb{Z}_{v}$.
  As an application of this result, given a positive integer $t\not\equiv
0\pmod{4}$, we prove that there are, for infinitely many pairs of $v$ and $k$,
at least $(1-o(1)) \frac{(\frac{v-t}{2})!}{\phi(v)} $ non-isomorphic
biembeddings of $K_{\frac{v}{t}\times t}$ whose face lengths are multiples of
$k$. Here $\phi(\cdot)$ denotes the Euler's totient function. Moreover, in case
$t=1$ and $v$ is a prime, almost all these embeddings define faces that are all
of the same length $kv$, i.e. we have a more than exponential number of
non-isomorphic $kv$-gonal biembeddings of $K_{v}$.
