Embeddings of mapping tori for end-periodic graph maps

TL;DR

This paper explores the properties of end-periodic homotopy equivalences on infinite graphs with finitely many ends, showing their mapping tori can be simplified to finite 2-complexes and embedded into finite-rank graph mapping tori.

Contribution

It establishes that end-periodic homotopy equivalences on infinite graphs have mapping tori that are homotopy equivalent to finite complexes and can embed into finite-rank graph mapping tori.

Findings

Mapping tori admit flowline-preserving homotopy equivalences with finite 2-complexes.

Under certain conditions, these tori embed into mapping tori of finite-rank graphs.

Every mapping class from an end-periodic homotopy equivalence has a representative with such an embedding.

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.