Two results on end spaces of infinite type surfaces

TL;DR

This paper investigates the topology of end spaces of infinite type surfaces, providing new examples and clarifying the relation between end types and local homeomorphism, advancing understanding in geometric topology.

Contribution

It presents novel examples of infinite type surfaces with specific end space properties and establishes an equivalence relation on end types aligning with local homeomorphism.

Findings

Existence of infinite type surfaces with non-self-similar end spaces but a unique maximal end type

The local complexity relation on end types is an equivalence relation

This relation coincides with the notion of local homeomorphism of end spaces

Abstract

We answer two questions about the topology of end spaces of infinite type surfaces and the action of the mapping class group that have appeared in the literature. First, we give examples of infinite type surfaces with end spaces that are not self-similar, but a unique maximal type of end, either a singleton or Cantor set. Secondly, we use an argument of Tsankov to show that the "local complexity" relation $\preccurlyeq$ on end types gives an equivalence relation that agrees with the notion of being locally homeomorphic.