Mapping tori of end-periodic graph maps (Adam Smith)

Abstract: Traditionally, the term end-periodic refers to a class of homeomorphism from an infinite-type surface to itself whose repeated iteration results in attracting/repelling dynamics near the ends of its domain. The mapping tori of such automorphisms, which arise naturally in the study of foliations on compact 3-manifolds, are by now part of a mature theory whose history spans several decades. And while much of this theory is couched in the language of manifolds, the fact that end-periodicity itself is just a dynamical phenomenon allows us to extend the notion of an end-periodic map to settings where these tools are not available.

In this talk, we will focus on one such generalization: end-periodic homotopy equivalences of infinite, locally finite graphs. Motivated by an interest in how their mapping tori offer a dynamical perspective on certain finitely generated free-by-cyclic groups, we will demonstrate how constructions involving classical end-periodic mapping tori can be adapted to this more combinatorial context. The group-theoretic consequences of these constructions will also be discussed.