Two prevailing theories for explaining social group or community structure are cohesion and identity. The social cohesion approach posits that social groups arise out of an aggregation of individuals that have mutual interpersonal attraction as they share common characteristics. These characteristics can range from common interests to kinship ties and from social values to ethnic backgrounds. In contrast, the social identity approach posits that an individual is likely to join a group based on an intrinsic self-evaluation at a cognitive or perceptual level. In other words group members typically share an awareness of a common category membership. In this work we seek to understand the role of these two contrasting theories in explaining the behavior and stability of social communities in Twitter. A specific focal point of our work is to understand the role of these theories in disparate contexts ranging from disaster response to socio-political activism. We extract social identity and social cohesion features-of-interest for large scale datasets of five real-world events and examine the effectiveness of such features in capturing behavioral characteristics and the stability of groups. We also propose a novel measure of social group sustainability based on the divergence in group discussion. Our main findings are: 1) Sharing of social identities (especially physical location) among group members has a positive impact on group sustainability, 2) Structural cohesion (represented by high group density and low average shortest path length) is a strong indicator of group sustainability, and 3) Event characteristics play a role in shaping group sustainability, as social groups in transient events behave differently from groups in events that last longer.