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Fixed Subgroups are Compressed in Free Groups

Abstract In this paper, we prove that the fixed subgroup of an arbitrary family of endomorphisms ψ i , i ∈ I, of a finitely generated free group F, is F-super-compressed. In particular, r(∩ i∈I Fix ψ i ) ≤ r(M) for every subgroup M ≤ F containing ∩ i∈I Fix ψ i . 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 Levitt.