Geometry of the graphs of nonseparating curves: covers and boundaries

TL;DR

This paper explores the geometric properties of nonseparating curve graphs on infinite type surfaces, focusing on covers, quasiconvexity, and boundary identification, and introduces new tools for their analysis.

Contribution

It demonstrates that lifts of nonseparating curves form quasiconvex subgraphs and reestablishes boundary identification results, introducing tools for studying infinite type surfaces.

Findings

Lifts of nonseparating curves are quasiconvex in covers.

The nonseparating curve graph has infinite diameter and is Gromov hyperbolic.

Boundary identified with ending laminations on subsurfaces.

Abstract

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparating curves to the nonseparating curve graph of the cover span quasiconvex subgraphs which are infinite diameter and not coarsely equal to the nonseparating curve graph of the cover. In the finite type case, we also reprove a theorem of Hamenst\"{a}dt identifying the Gromov boundary with the space of ending laminations on full genus subsurfaces. We introduce several tools based around the analysis of bicorn curves and laminations which may be of independent interest for studying the geometry of nonseparating curve graphs of infinite type surfaces and their boundaries.