Convexity of strata in diagonal pants graphs of surfaces

TL;DR

This paper investigates the convexity properties of strata within the diagonal pants graph of a surface, revealing convex flat subgraphs of all ranks, and draws parallels with Weil-Petersson geometry.

Contribution

It establishes convexity results for strata in the diagonal pants graph, introducing new geometric insights and identifying convex flat subgraphs of all ranks.

Findings

Convexity of strata in the diagonal pants graph proven.

Existence of convex flat subgraphs of every possible rank.

Analogies with Weil-Petersson completion properties.

Abstract

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.