On Equalizers of Sections

LetGandMbe groups, anda, b:G → G * Mgroup-theoretic sections of the natural projectionG * M → G. We use the Almost Stability Theorem, pro-trees, and new folding sequence techniques to show that ifGis finitely generated, then the equalizer ofaandbis a free factor ofG, which answers a question of G. M. Bergman.