The Equalizer Conjecture For The Free Group of Rank Two

Abstract

Abstract The equalizer of a set of homomorphisms $S: F(a, b)\rightarrow F(\Delta)$ has rank at most two if S contains an injective map and is not finitely generated otherwise. This proves a strong form of Stallings’ Equalizer Conjecture for the free group of rank two. Results are also obtained for pairs of homomorphisms $g, h:F(\Sigma)\rightarrow F(\Delta)$ when the images are inert in, or retracts of, $F(\Delta)$.