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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo fence="true" lspace="0em" rspace="0em">∣</m:mo> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mi>y</m:mi> <m:mi>q</m:mi> </m:msup> </m:mrow> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> G_{p,q} . Our main tool is an action of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{Aut}(G_{p,q}) on the Bass–Serre tree of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> G_{p,q} which is compatible with the original action, in the sense that it extends the original action of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> G_{p,q} on its Bass–Serre tree.