Tsachik Gelander of Northwestern University presents a research seminar talk given at the Joint IAS/Princeton University Groups and Dynamics Seminar at the Institute for Advanced Study. The lecture opens with Weigold's conjecture, which claims that for a finite simple group G and n greater than 2, the product replacement graph on n-generating tuples is connected, a problem still open and known to imply the graph is an expander once n exceeds 3. Gelander then turns to the analogous question when G is a compact Lie group rather than a finite simple group, walking through what partial results are actually known, where the arguments break down, and which stronger conjectures he believes might be true but cannot currently prove. The talk is aimed at specialists in group theory and dynamics, moving through definitions quickly before settling into the open problems that motivate the seminar's ongoing research discussion.