  In this paper, we shall prove that an ultraproduct of non-abelian finite
simple groups is either finite simple, or has no finite dimensional unitary
representation other than the trivial one. Then we shall generalize this result
for other kinds of quasirandom groups. A group is called D- quasirandom if all
of its nontrivial representations over the complex numbers have dimensions at
least D. We shall study the question of whether a non-principal ultraproduct of
a given sequence of quasirandom groups remains quasirandom, and whether an
ultraproduct of increasingly quasirandom groups becomes minimally almost
periodic (i.e. no non-trivial finite-dimensional unitary representation at
all). We answer this question in the affirmative when the groups in question
are simple, quasisimple, semisimple, or when the groups in question have
bounded number of conjugacy classes in their cosocles (the intersection of all
maximal normal subgroups), or when the groups are arbitrary products (not
necessarily finite) of the groups just listed. We shall also present with an
ultraproduct of increasingly quasirandom groups with a non-trivial
one-dimensional representation. We also obtain some results in the case of
semi-direct products and short exact sequences of quasirandom groups. Finally,
two applications of our results are given, one in triangle patterns of
quasirandom groups and one in self-Bohrifying groups. Our main tools are some
variations of the covering number for groups, different kinds of length
functions on groups, and the classification of finite simple groups.
