Commutator formulas in groups of Kac-Moody type

In this thesis, I prove that root groups associated to nested roots in certain rank 3 hyperbolic buildings must commute. Also, I give necessary and sufficient conditions for existence of nontrivial commutators between root groups associated to roots generating infinite root systems in Kac-Moody groups. Lastly, a gap in the literature regarding “reduction to rank 2” in Kac-Moody root systems is filled.