Narrow Results

Let G be the fundamental group of a graph of groups with finite edge groups and f an endomorphism of G. We prove a structure theorem for the subgroup Fix(f), which consists of the elements of G fixed by f, in the case where the endomorphism f of G maps vertex groups into conjugates of themselves.

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).