We study online social group dynamics based on how group members diverge in their online discussions. Previous studies mostly focused on the link structure to characterize social group dynamics, whereas the group behavior of content generation in discussions is not well understood. Particularly, we use Jensen-Shannon (JS) divergence to measure the divergence of topics in user-generated contents, and how it progresses over time. We study Twitter messages (tweets) in multiple real-world events (natural disasters and social activism) with different times and demographics. We also model structural and user features with guidance from two socio-psychological theories, social cohesion and social identity, to learn their implications on group discussion divergence. Those features show significant correlation with group discussion divergence. By leveraging them we are able to construct a classifier to predict the future increase or decrease in group discussion divergence, which achieves an area under the curve (AUC) of 0.84 and an F-1 score (harmonic mean of precision and recall) of 0.8. Our approach allows to systematically study collective diverging group behavior independent of group formation design. It can help to prioritize whom to engage with in communities for specific topics of needs during disaster response coordination, and for specific concerns and advocacy in the brand management.