Embeddings of mapping tori for end-periodic graph maps

Abstract

End-periodic homotopy equivalences of infinite, locally finite graphs serve as dimension-one analogs of the end-periodic automorphisms traditionally defined on infinite-type surfaces. We demonstrate that if $\Gamma$ is an infinite graph with finitely many ends, and $g \colon \Gamma \to \Gamma$ is end-periodic, then its mapping torus $Z_g$ admits a flowline-preserving homotopy equivalence with a finite 2-complex. With additional hypotheses on $g$, this compactified mapping torus subsequently embeds in the mapping torus of a homotopy equivalence on a finite-rank graph via a $\pi_1$-injective, flow-preserving map. We prove that every mapping class of $\Gamma$ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.