Members of thin \Pi_1^0 classes and generic degrees.

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 \mathbf {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 \mathbf {0}' contain members of thin \Pi ^{0}_{1} classes. [ABSTRACT FROM AUTHOR]