The equalizer conjecture for the free group of rank two

Abstract

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

arXiv comment19 pages. v2 incorporates minor edits (mostly typos). v3 is the final version accepted for publication, with improved structural and algorithmic results for retracts