Fixed subgroups are compressed in surface groups

For a compact surface Σ (orientable or not, and with boundary or not), we show that the fixed subgroup, Fix ℬ, of any family ℬ of endomorphisms of π1(Σ) is compressed in π1(Σ), i.e. rk ( Fix ℬ) ≤ rk (H) for any subgroup Fix ℬ ≤ H ≤ π1(Σ). On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, G, of finitely many free and surface groups, and give a characterization of when G satisfies that rk ( Fix ϕ) ≤ rk (G) for every ϕ ∈ Aut (G).