Retracts of free groups and a question of Bergman

Ilir Snopce, Slobodan Tanushevski, Pavel Zalesskii

Let $F_n$ be a free group of finite rank $n\ge 2$. We prove that if $H$ is a subgroup of $F_n$ with $\text{rk}(H)=2$ and $R$ is a retract of $F_n$, then $H\cap R$ is a retract…