Event-based cameras generate asynchronous streams of events, triggered proportionally to the logarithmic change of brightness in the scene. These cameras have very low latency and high dynamic range suitable to address challenging motion scenarios in robotics. In this work, we explore a new event-based line-SLAM approach following a parallel tracking and mapping philosophy. Our fast tracking algorithm, produces accurate camera pose estimates at a high rate by minimizing the event-line reprojection error with an error-state Kalman filter formulated entirely with Lie theory. The mapping thread leverages the natural edge highlighting strength of events to recover and optimize straight lines in human-made scenarios. The proper manipulation of matrix sparsity as well as the information sharing between tracking and mapping nodes allow us to achieve real-time performance on a standard multi-core CPU. This system was tested on several scenarios rich in straight edge objects, and compared against, ground truth and frame and event based state-of-the-art approaches.