Strip maps of small surfaces are convex

TL;DR

This paper proves that the strip map, which relates to infinitesimal deformations of small bordered hyperbolic surfaces, forms a convex hypersurface for surfaces like the punctured torus or thrice punctured sphere.

Contribution

It establishes the convexity of the strip map's image specifically for small complexity surfaces such as the punctured torus and thrice punctured sphere.

Findings

The image of the strip map is a convex hypersurface for small surfaces.

Convexity holds for surfaces of small complexity like the punctured torus.

The result applies to the arc complex and deformation space of hyperbolic surfaces.

Abstract

The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of small complexity: the punctured torus or thrice punctured sphere.