Relative train tracks and generalized endperiodic graph maps

Abstract

Motivated by the work of Cantwell–Conlon–Fenley on endperiodic homeomorphisms of infinite‐type surfaces, we define and study endperiodic and generalized endperiodic maps of an infinite graph with finitely many ends. Adapting the work of Bestvina–Handel to the infinite‐type setting, we define endperiodic relative train track maps. We prove that any generalized endperiodic map is homotopic to a generalized endperiodic relative train track map, via a combinatorially bounded homotopy equivalence.

We prove three applications of our main theorem. We first show that the (largest) Perron–Frobenius eigenvalue of a relative train track representation of a generalized endperiodic map is a canonical quantity associated with as it admits a canonical group theoretic interpretation. Moreover, the (largest) Perron–Frobenius eigenvalue and the topological entropy of a relative train track map is the smallest among its proper homotopy equivalence class. Furthermore, for endperiodic homeomorphisms of fibered surfaces we show that the relative train track maps provide a way to obtain the Handel–Miller laminations.