The non-peripheral curve graph and divergence in big mapping class groups

Abstract

We introduce a numerical invariant $ζ(Σ)$ measuring the end-complexity of $Σ$ and use it to organize coarse-geometric features of Map($Σ$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(Σ)$, whose vertices are those essential simple closed curves that cannot be pushed out of every compact subsurface, with edges given by disjointness. Assuming Map($Σ$) is CB-generated and $ζ(Σ)\ge 5$, we prove that $C_{\rm np}(Σ)$ is connected, has infinite diameter, is Gromov hyperbolic, and that the Map($Σ$)-action has unbounded orbits. As applications, we show that if $ζ(Σ)\ge 4$ then Map($Σ$) has infinite coarse rank, and if $ζ(Σ)\ge 5$ then Map($Σ$) has at most quadratic divergence, hence is one-ended.

Figures (3)

• Figure 1: A surface $\Sigma$ with $\zeta(\Sigma) = 5$.
• Figure 2: When $\zeta(\Sigma) = 4$ the graph ${C_{\rm np}}$ may not be connected. But we can still find arbitrarily large number of disjoint non-peripheral curves.
• Figure 3: Chain of flats form alternating product region

Theorems & Definitions (87)

• Theorem A: Mapping class groups with infinite coarse rank
• Conjecture B
• Theorem C: Quadratic divergence bound
• Corollary D
• Theorem E: Geometry of ${C_{\rm np}}(\Sigma)$
• Definition 2.1
• Theorem 2.2: Rosendal
• Definition 2.3
• Example 2.4
• Theorem 2.5: MR, Theorem 1.6