Iwip endomorphisms of free groups and fixed points of graph selfmaps

Abstract

In a paper from 2011, Jiang, Wang and Zhang studied the fixed points and fixed subgroups of selfmaps on a connected finite graph or a connected compact hyperbolic surface 𝑋. In particular, for any selfmap f : X → X f\colon X\to X , they proved that a certain quantity defined in terms of the characteristic chr ⁡ ( f , F ) \operatorname{chr}(f,\mathbf{F}) and the index ind ⁡ ( f , F ) \operatorname{ind}(f,\mathbf{F}) of a fixed point class 𝐅 of 𝑓 is bounded below by 2 ⁢ χ ⁢ ( X ) 2\chi(X) , where χ ⁢ ( X ) \chi(X) is the Euler characteristic of 𝑋. In this paper, we give a sufficient condition for when equality holds and hence we partially answer a question of Jiang. We do this by studying iwip outer endomorphisms of free groups acting on stable trees.