Hugo Parlier

Collars and partitions of hyperbolic cone-surfaces

For compact Riemann surfaces, the collar theorem and Bers’ partition theorem are major tools for working with simple closed geodesics. The main goal of this article is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone angles to be strictly less than π to be able to consider partitions.

Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmuller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conjecture.

Constructing convex planes in the pants complex

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.

Convexity of strata in diagonal pants graphs of surfaces

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.

Kissing numbers for surfaces

The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus

A geometric characterization of orientation-reversing involutions

g

On Harnack’s theorem and extensions: A geometric proof and applications

can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice sphere packings, are known to grow, as a function of genus, at least like

A short note on short pants

g^{\sfrac{4}{3}-\epsilon}

Geometry of Riemann Surfaces: Simple closed geodesics of equal length on a torus

for any

Systoles and kissing numbers of finite area hyperbolic surfaces

\epsilon >0