Hyperbolicity of the graph of non-separating multicurves

TL;DR

This paper proves that the graph of non-separating multicurves with fewer than g/2+1 components on a surface of genus g is hyperbolic, revealing geometric properties of these combinatorial structures.

Contribution

It establishes the hyperbolicity of the non-separating multicurve graph for certain values of k, extending understanding of its geometric structure.

Findings

The graph is hyperbolic when k < g/2+1.

Provides new insights into the geometry of multicurve graphs.

Extends previous results on curve complexes.

Abstract

A non-separating multicurve of a surface S of genus g with m punctures is a multicurve c so that S-c is connected. For k>0 define the graph of non-separting k-multicurves to be the graph whose vertices are non-separating multicurves with k components and where two such multicurves are connected by an edge if they can be realized disjointly. We show that if k is smaller than g/2+1 then this graph is hyperbolic.