Economical separability in free groups

Abstract

Consider the rank n free group Fn with basis X. Bogopol’skiĭ conjectured in [1, Problem 15.35] that each element w ∈ Fn of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ Fn of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of Fn, as well as the separability by a subgroup of index at most |w|/2 + 2 in general.