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, FixB, of any family B of endomorphisms of π1(Σ) is compressed in π1(Σ), i.e., rk(FixB) 6 rk(H) for any subgroup FixB 6 H 6 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(Fixφ) 6 rk(G) for every φ ∈ Aut(G).