Uniform local finiteness of the curve graph via subsurface projections

Abstract

The curve graphs are not locally finite. In this paper, we show that the curve graphs satisfy a property which is equivalent to graphs being uniformly locally finite via Masur--Minsky's subsurface projections. As a direct application of this study, we show that there exist computable bounds for Bowditch's slices on tight geodesics, which depend only on the surface. As an extension of this application, we define a new class of geodesics, weak tight geodesics, and we also obtain a computable finiteness statement on the cardinalities of the slices on weak tight geodesics.

25 pages. Minor revisions following comments from the referees. Bibliography updated. No mathematical change from v3

References in corpus (3)

• Uniform hyperbolicity of the curve graph via surgery sequences
• Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs
• A short proof of the bounded geodesic image theorem

Cited by in corpus (2)

• Intersection numbers in the curve graph with a uniform constant
• Intersection numbers in the curve complex via subsurface projections