# Constructing convex planes in the pants complex

@inproceedings{Aramayona2007ConstructingCP, title={Constructing convex planes in the pants complex}, author={Javier Aramayona and Hugo Parlier and Kenneth J. Shackleton}, year={2007} }

Our main theorem identifies a class of totally geodesic subgraphs of the 1-skeleton of the pants complex, referred to as the pants graph, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the pants graph of any surface of complexity at least 3.

#### 8 Citations

Convexity of strata in diagonal pants graphs of surfaces

- Mathematics
- 2011

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… Expand

Products of Farey graphs are totally geodesic in the pants graph

- Mathematics
- 2013

We show that for a surface Σ, the subgraph of the pants graph determined by fixing a collection of curves that cut Σ into pairs of pants, once-punctured tori, and four-times-punctured spheres is… Expand

Geodesic axes in the pants complex of the five-holed sphere

- Mathematics
- 2011

Abstract We study the synthetic geometry of the pants graph of the 5-holed sphere, establishing the existence of geodesics connecting any vertex or ideal point to any ideal point. We prove the… Expand

Large flats in the pants graph

- Mathematics
- 2013

This note is about the geometry of the pants graph P(S), a natural simplicial graph associated to a finite type topological surface S where vertices represents pants decompositions. The main result… Expand

Geometric simplicial embeddings of arc-type graphs

- Mathematics
- 2019

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied… Expand

The geometry of flip graphs and mapping class groups

- Mathematics
- Transactions of the American Mathematical Society
- 2019

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a… Expand

Hyperbolicity of the genus two Hatcher–Thurston complex

- Mathematics
- 2013

For the genus 1 surface with n punctures F1,n, we show that the Hatcher–Thurston complex $${\fancyscript{HT}(F_{1,n})}$$ is hyperbolic. For the genus 2 closed surface F2, we show that the… Expand

Combinatorial methods in Teichmüller theory

- Mathematics
- 2013

Dans cette these nous etudions certains proprietes combinatoires et geometriques des complexes d'arcs des surfaces de type fini. Nous demontrons que le groupe d'automorphisme du complexe d'arcs est… Expand

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In recent work of Brock's, the pants graph is shown to be a combinatorial model for the completion of the Weil-Petersson metric on Teichmuller space. We prove that every Farey graph embedded in the… Expand

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<abstract abstract-type="TeX"><p>Let <i>S</i> be a surface with genus <i>g</i> and <i>n</i> boundary components, and let <i>d(S)</i> = 3<i>g</i> - 3 + <i>n</i> denote the number of curves in any… Expand

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