An investigation of the combinatorics of square-tiled surfaces, and the geometry of Teichmüller space

The context of the research: The broad area of research is geometry, topology, mathematical
analysis, and algebra. The specific context is hyperbolic geometry, geometric group theory,
mapping class groups, and Teichmuller theory. The primary project focusses on constructing
square-tiled surfaces with specific combinatorial structure. The secondary project aims to
investigate the hyperbolic-like geometry of Teichmuller space; specifically, the notion of statistical
hyperbolicity. Successful completion of the projects will add fresh entries to the literature on the
constructions and applications of square-tiled surfaces, and on the geometry of Teichmuller
space. This is fundamental research, and has potential long-term impact in any other scientific
fields where surfaces come into play including physics, chemistry, and biology/medicine.
Aims and objectives: The aim of the primary project is to provide minimal constructions of squaretiled
surfaces that have, simultaneously, a single horizontal cylinder and a single vertical cylinder.
Moreover, such surfaces are to be constructed in every connected component of every stratum of
the moduli space of translation surfaces. Constructions of surfaces satisfying the horizontal
cylinder condition were given by Zorich, but there is no obvious way to extend these
constructions to our setting. The surfaces we intend to construct will have applications to the
study of the coarse-geometry of Teichmuller space. The aim of the secondary project is to prove
that Teichmuller space is statistically hyperbolic with respect to a harmonic measure arising from
a random walk in the mapping class group. The notion of statistical hyperbolicity, introduced by
Duchin-Lelièvre-Mooney, encapsulates whether a space is on average hyperbolic at large scales.
Dowdall-Duchin-Masur prove that Teichmuller space is statistically hyperbolic with respect to
many Lebesgue-class measures, but such measures are known by Gadre and Gadre-Maher-
Tiozzo to be singular with respect to the harmonic measures considered above, and so different
tools are required in our setting.
Novelty of the research methodology: For the primary project, the main new idea is to approach
the constructions within the setting of filling curves on surfaces. It is expected that this framing of
the problem should allow one to have a better handle on the combinatorics of the constructions.
The main new idea in the secondary project is to utilise the geodesic statistics considered in the
proofs of the singularity of measure. More precisely, it is expected that typical geodesics with
respect to harmonic measures should behave more hyperbolic-like. The project aims to make this
precise and then feed these statistics into the criterion provided by Dowdall-Duchin-Masur.
Alignment to EPSRC's strategies and research area: The project is intra-disciplinary. It pushes
the boundaries of the interaction between geometry, topology, mathematical analysis, and
algebra, incorporating techniques from all of these areas.