Translation surfaces with large systoles

Abstract

In this paper we continue to investigate the systolic landscape of translation surfaces started in [CHMW]. We show that there is an infinite sequence of surfaces $(S_{g_k})_k$ of genus $g_k$, where $g_k \to \infty$ with large systoles. On the other hand we show that for hyperelliptic surfaces we can find a suitable homology basis, where a large number of loops that induce the basis are short.

Figures (5)

• Figure 1: Construction of a rectangle $R$ with a slit of length $L$. Dashed lines are glued together.
• Figure 2: Dashed and full lines are identified in each of the two rectangles to obtain the two tori $P_1$ and $P_2$. These are glued together along the slit to obtain the surface $X$.
• Figure 3: Construction of a torus $T$ with $2L^2$ slits from copies of the rectangle $R$.
• Figure 4: An arc $c$ of the geodesic $\gamma$ connecting two cone points $p$ and $q$.
• Figure 7: A hyperelliptic surface $S$ of genus four with ten Weierstrass points.

Theorems & Definitions (17)

• Theorem 1.1: chms
• Theorem 1.2: Surfaces with large systoles
• Theorem 1.3: Hyperelliptic surfaces
• Theorem 2.1: Surfaces with large systoles
• Lemma 2.2
• proof : Proof of Lemma \ref{['lem:sys2L']}
• Lemma 2.5
• proof
• Theorem 3.1: Homology systoles of hyperelliptic surfaces
• proof