Fixed subgroups are compressed in free groups

In this paper we prove that the fixed subgroup of an arbitrary family of endomorphisms ψi, i ∈ I, of a finitely generated free group F , is F -super-compressed. In particular, r(∩i∈IFix ψi) ≤ r(M) for every subgroup M ≤ F containing ∩i∈IFix ψi. This provides new evidence towards the inertia conjecture for fixed subgroups of free groups. As a corollary, we show that, in the free group of rank n, every strictly ascending chain of fixed subgroups has length at most 2n. This answers a question of G. Levitt.