In this talk, we will present certain rigidity results for group actions on compact spaces. In the first part, we will provide a new characterization of one end groups via cocycle superrigidity of their full shifts. As a consequence, we have an application in continuous orbit equivalence rigidity. In the second part, we prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property then it is C0 local rigid. A new characterization of subshifts of finite type over finitely generated groups in term of pseudo-orbit tracing property is also mentioned. The first part is joint with Yongle Jiang and the second part is joint work with Keonhee Lee.