Fixed points of automorphisms of torus knot groups

Abstract We completely classify fixed point subgroups in torus knot groups, that is, groups of the form G p , q = ⟨ x , y ∣ x p = y q ⟩ G_{p,q}=\langle x,y\mid x^{p}=y^{q}\rangle . We not only give the isomorphism type, but also the explicit generators for the fixed point subgroup of each automorphism of G p , q G_{p,q} . Our main tool is an action of Aut ⁢ ( G p , q ) \mathrm{Aut}(G_{p,q}) on the Bass–Serre tree of G p , q G_{p,q} which is compatible with the original action, in the sense that it extends the original action of G p , q G_{p,q} on its Bass–Serre tree.