arXiv subjects

Achintya Dey

Publications and source records attributed to Achintya Dey.

5 recordsLinked to original sources

arXiv2026

Systolic Geometry and Combinatorics of Surfaces

This thesis investigates two themes. The first studies systolic embeddings of graph on translation surfaces, proving existence results for essential and cellular systolic embeddings and determining extremal genera for certain graph families. The second examines hyperbolic surfaces in the Thurston set, constructing a one-parameter deformation that increases the sys- tole length and produces Bolza-like surfaces, i.e., surfaces whose systolic geodesics decompose the surfaces into triangles, for infinitely many genera. A symmetry property of geodesic inter- sections on these surfaces is also established, together with an application of these Bolza-like surfaces.

math.GT↗

arXiv2026

Short homology bases for translation surfaces

Let $S$ be a closed translation surface of genus $g\ge 2$ with $\mathop{area}(S)= 4πg$. We show that for any $λ\in (0, 1)$ there exist $\lfloor λ\cdot g\rfloor$ homologically independent simple closed curves of length at most $ C(λ) \cdot \log(g)$, where $C(λ)$ is a constant that depends only on $λ$. This result is obtained from a mixed graph construction based on a Voronoi graph of the surface with the cone points as seeds and its dual graph. We also give a complementary result using only the Voronoi graph that produces $2g$ short loops that form a homology basis. This construction, however, depends on the length of a shortest saddle connection of the surface $S$.

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arXiv2026

Systole Increasing Deformations to Maximal Translation Surfaces

A unit-area translation surface is called \emph{maximal} if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a deformation does not exist in general due to the existence of local but not global maxima of the systole function, we prove that it exists for square-tiled surfaces and translation surfaces obtained from regular hexagons and regular octagons. In each case, we explicitly construct a continuous deformation to a maximal translation surface along which the systole is strictly increasing. Finally, the preceding construction yields another family of translation surfaces admitting continuous systole-increasing deformations to maximal surfaces.

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arXiv2026

Bolza-like surfaces in the Thurston set

A surface in the Teichmüller space, where the systole function admits its maximum, is called a maximal surface. For genus two, a unique maximal surface exists, which is called the Bolza surface, whose systolic geodesics give a triangulation of the surface. We define a surface as Bolza-like if its systolic geodesics decompose the surface into $(p, q, r)$-triangles for some integers $p,q,r$. In this article, we will provide a construction of Bolza-like surfaces for infinitely many genera $g\geq 9$. Next, we see an intriguing application of Bolza-like surfaces. In particular, we construct global maximal surfaces using these Bolza-like surfaces. Furthermore, we study a symmetric property satisfied by the systolic geodesics of our Bolza-like surfaces. We show that any simple closed geodesic intersects the systolic geodesics at an even number of points.

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arXiv2025

Systolic embedding of graphs on translation surfaces

An embedding of a graph on a translation surface is said to be \emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation surface. The embedding is said to be \emph{cellular} (respectively \emph{essential}) if each complementary region is a topological disk (respectively not a topological disk). In this article, we prove that any finite graph admits an essential-systolic embedding on a translation surface and estimate the genera of such surfaces. For a wedge $Σ_n$ of $n$ circles, $n\geq2$, we investigate that $Σ_n$ admits cellular-systolic embedding on a translation surface and compute the minimum and maximum genera of such surfaces. Finally, we have identified another rich collection of graphs with more than one vertex that also admit cellular-sytolic embedding on translation surfaces.

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