Large flats in large subgraphs of fine curve graphs

TL;DR

This paper investigates large subgraphs of fine curve graphs related to surfaces, showing they contain flats of all finite dimensions and computing bounds on distances within certain fibers.

Contribution

It demonstrates that specific large subgraphs of fine curve graphs are not hyperbolic and contains flats of every finite dimension, expanding understanding of their geometric properties.

Findings

Large subgraphs contain flats of all finite dimensions.

Certain fibers over vertices are not hyperbolic.

Bounds on distances in single-isotopy-class fine curve graphs are provided.

Abstract

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves and whose edges connect disjoint curves. Following a rich history of hyperbolicity of various graphs associated to surfaces, the fine curve graph was shown to be hyperbolic by Bowden-Hensel-Webb, while the curve graph, obtained from the fine curve graph by collapsing subgraphs corresponding to isotopy classes, was first proven to be hyperbolic by Masur-Minsky. We show that certain large subgraphs of fine curve graphs, including fibers over a vertex of the curve graph, are not hyperbolic. Indeed, such graphs contain flats of every finite dimension. We then compute bounds on distances in fibers over a vertex of the curve graph, which we call single-isotopy-class fine curve graphs.