10 oct 2024 -- 11:00
Aula Riunioni, Dipartimento di Matematica, Univesità di Pisa
Abstract.
If $G$ is a connected finite graph (i.e., $G$ has finitely many vertices and edges), a graph map (i.e., a homotopy equivalence) $f: G \to G$ can be viewed as a topological representative of an outer automorphism of the finitely generated free group $\pi_1G$. The outer automorphism group $Out(F_n)$ of a free group $F_n$ on n generators has been proven to share many similar properties of the mapping class group of a compact surface.
If $G$ is an infinite graph (i.e.,$G$ has infinitely many vertices and edges), the structure of a (proper) graph map $f: G \to G$ is more complicated and less understood. Inspired by the recent work of Cantwell-Conlon-Fenley on homeomorphisms of infinite type surfaces, we introduce and study endperiodic graph maps $f: G \to G$ where $G$ has finitely many ends. We prove that any such a map is homotopic to an endperiodic relative train track map, which is a normal form for an endperiodic graph map. Moreover, we show that the Perron-Frobenius eigenvalue of the transition matrix is a canonical quantity associated to the map. This is joint work with Chenxi Wu.