Georgia Institute of Technology College of Sciences

In this talk, I will present a geometric algorithm for determining whether a given set of elements in SO+(n,1) generates a discrete subgroup, as well as identifying the relators for the corresponding group presentation. The algorithm constructs certain hyperbolic manifolds that are always complete, a key condition for applying Poincaré Fundamental Polyhedron Theorem and ensuring the algorithm is valid. I will also introduce a generalization of this algorithm to the Lie group SL(n, R) and explore how the completeness condition extends to this broader setting.