断绝 implies a complete and deliberate severing — it's final and absolute. Common collocations: 断绝关系 (sever relations), 断绝联系 (cut off contact), 断绝来往 (cut off dealings).
grammar
断绝 is typically followed by the object being cut off: 断绝 + [关系/联系/来往/希望]. It can also be used in passive: 被断绝.