In this talk, I will discuss the geodesic completeness problem for left-invariant metrics on Lie groups from several complementary perspectives. On the one hand, fixing a metric Lie group \( (G,g) \), geodesic completeness can be studied via the associated Euler-Arnold equation on the Lie algebra. In low dimensions, this dynamical viewpoint allows for a detailed analysis, leading in particular to classification results in dimension three. On the other hand, fixing a Lie group \( G \), it is natural to ask whether it admits incomplete left-invariant metrics, or conversely, whether all such metrics are necessarily complete.

A recent result by Elshafei, Ferreira, Sánchez, and Zeghib shows that a certain class of Lie groups forces completeness for every left-invariant semi-Riemannian metric. In a dual ongoing joint work with Ana Cristina Ferreira, we investigate Lie-algebraic obstructions to the completeness of all left-invariant metrics, aiming to understand this problem from a purely algebraic viewpoint.

Finally, I will briefly discuss a related notion of semicompleteness for left-invariant holomorphic Riemannian metrics on complex Lie groups, and explain how unimodularity in dimension three plays a stabilizing role in this setting.

The talk is based on joint work with Ana Cristina Ferreira and Abdelghani Zeghib, as well as work in progress.