  A $\Pi^{0}_{1}$ class $P$ is thin if every $\Pi^{0}_{1}$ subclass $Q$ of $P$
is the intersection of $P$ with some clopen set. In 1993, Cenzer, Downey,
Jockusch and Shore initiated the study of Turing degrees of members of thin
$\Pi^{0}_{1}$ classes, and proved that degrees containing no members of thin
$\Pi^{0}_{1}$ classes can be recursively enumerable, and can be minimal degree
below {\bf 0}$'$. In this paper, we work on this topic in terms of genericity,
and prove that all 2-generic degrees contain no members of thin $\Pi^{0}_{1}$
classes. In contrast to this, we show that all 1-generic degrees below {\bf
0}$'$ contain members of thin $\Pi^{0}_{1}$ classes.
