The maximum number of systoles for genus two Riemann surfaces with abelian differentials

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,\omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $\omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.Comment: 41 page

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus two surface and that the bound is realized by a unique translation surface up to homothety. We also provide sharp upper bounds on the the number of homotopy classes of systoles for a holomorphic $1$-form with a single zero in terms of the genus

Systolic geometry of translation surfaces

Let $S$ be a translation surface of genus $g > 1$ with $n$ cone points $(p_i)_{i=1,\ldots,n}$ with cone angle $2\pi \cdot (k_i+1)$ at $p_i$, where $k_i \in \mathbb{N}$. In this paper we investigate the systolic landscape of these translation surfaces for fixed genus.Comment: 25 pages, 4 figures. Added explicit computations of systoles in the graph of saddle connections for origamis in H(1,1) and a criterion to decide whether such systoles define systoles on the translation surfac

Well-rounded equivariant deformation retracts of Teichm\"uller spaces

In this paper, we construct spines, i.e., \Mod_g-equivariant deformation retracts, of the Teichm\"uller space \T_g of compact Riemann surfaces of genus $g$. Specifically, we define a \Mod_g-stable subspace $S$ of positive codimension and construct an intrinsic \Mod_g-equivariant deformation retraction from \T_g to $S$. As an essential part of the proof, we construct a canonical \Mod_g-deformation retraction of the Teichm\"uller space \T_g to its thick part \T_g(\varepsilon) when $\varepsilon$ is sufficiently small. These equivariant deformation retracts of \T_g give cocompact models of the universal space \underline{E}\Mod_g for proper actions of the mapping class group \Mod_g. These deformation retractions of \T_g are motivated by the well-rounded deformation retraction of the space of lattices in $\R^n$. We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichm\"uller space.Comment: A revised version. L'Enseignement Mathematique, 201

Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs

In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichm\"uller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials. Finally, we present a further application to filling pairs on punctured surfaces by constructing filling pairs whose algebraic and geometric intersection numbers are equal.Comment: 39 pages, 25 figures. Final version incorporating changes suggested by the referee. To appear in Trans. Amer. Math. So

Investigating the cohomological dimensions of Mg

We discuss the problem of determining the de Rham, Dolbeault and algebraic cohomological dimension of the moduli space Mg of Riemann surfaces of genus g, focusing on possible strategies of attack and then concentrating on exhaustion functions. In the final section we explain how these techniques can be employed to provide a non-trivial upper bound for the Dolbeault cohomological dimension of Mg