Searcharxiv⌕ Search

arXiv subjects

Chandrika Sadanand

Publications and source records attributed to Chandrika Sadanand.

9 recordsLinked to original sources
arXiv2026

Straight-line trajectories on the Mucube

The dynamics of straight line flows on compact half-translation surfaces (surfaces formed by gluing Euclidean polygons edge-to-edge via translations possibly composed with rotation by $π$) has been widely studied due to their connections to polygonal billiards and Teichmüller theory. However, much less is known when the underlying surface is non-compact or infinite type. In this paper, we consider the straight line flow of the Mucube -- an infinite $\mathbb{Z}^3$-periodic half-translation square-tiled surface -- first written about by Coxeter and Petrie and more recently studied by Athreya--Lee and Gutiérrez-Romo--Lee--Sánchez. We give a geometric description of the flow's periodic and drift orbits in terms of the Mucube's rigid symmetries, and we give a complete characterization of the set of directions in which the straight line flow is periodic on the Mucube -- first in terms of a genus one quotient and second in terms of an infinitely generated subgroup of $\mathrm{SL}_2(\mathbb{Z})$. We use the latter characterization to obtain the Veech group (i.e. group of derivatives of affine diffeomorphisms) of the Mucube. Finally, we prove density of the sets of periodic and ergodic directions.

math.DS↗
arXiv2026

Realizing groups as symmetries of infinite translation surfaces

We provide a complete classification of groups that can be realized as isometry groups of a translation surface $M$ with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if $S$ has no non-displaceable subsurfaces and its space of ends is self-similar, then every countable subgroup of $\operatorname{GL}^+(2,\mathbb{R})$ can be realized as the Veech group of a translation surface $M$ homeomorphic to $S$. The latter result generalizes and improves upon the previous findings of Przytycki-Valdez-Weitze-Schmithüsen and Maluendas-Valdez. To prove these results, we adapt ideas from the work of Aougab-Patel-Vlamis, which focused on hyperbolic surfaces, to translation surfaces.

math.GT↗
arXiv2025

Veech groups of covers of the Chamanara surface

We study finite abelian covers of the Chamanara surface, an example of a finite-area infinite translation surface with interesting dynamics and a large Veech group. Specifically, the Veech group of the Chamanara surface is a virtually free group on two generators. We characterize when finite abelian covers have large Veech groups themselves, namely when their Veech group has finite index in that of the Chamanara surface. For degree-2 covers, we provide a detailed analysis of these finite-index Veech groups. As an application, we prove that every free group arises as the projective Veech group of a finite-area infinite translation surface.

math.GT↗
arXiv2025

Flip graph and arc complex finite rigidity

A subcomplex $\mathcal{X}$ of a cell complex $\mathcal{C}$ is called \emph{rigid} with respect to another cell complex $\mathcal{C}'$ if every injective simplicial map $λ:\mathcal{X} \rightarrow \mathcal{C}'$ has a unique extension to an injective simplicial map $ϕ:\mathcal{C}\rightarrow \mathcal{C}'$. We say that a cell complex exhibits \emph{finite rigidity} if it contains a finite rigid subcomplex. Given a surface with marked points, its \textit{flip graph} and \textit{arc complex} are simplicial complexes indexing the triangulations and the arcs between marked points, respectively. In this paper, we leverage the fact that the flip graph can be embedded in the arc complex as its dual to show that finite rigidity of the flip graph implies finite rigidity of the arc complex. Thus, a recent result of the second author on the finite rigidity of the flip graph implies finite rigidity of the arc complex for a broad class of surfaces. Notably, this includes surfaces with boundary -- a setting where finite rigidity of the arc complex was previously unknown.

math.GT↗
arXiv2023

Heegaard splittings and virtually special square complexes

We give a new perspective of Heegaard splittings in terms square complexes and Guirardel's notion of a \textit{core} which allows for combinatorial measurement of the obstruction to being a connect sum of Heegaard diagrams. A Heegaard splitting is a decomposition of a closed orientable $3$-manifold into two isomorphic handle bodies that have a shared boundary surface. Usually, a number of curves on the shared boundary surface, called a Heegaard diagram, are used to describe a Heegaard splitting. We define a larger object, the \textit{augmented Heegaard diagram}, by building on methods of Stallings and Guirardel to encode the information of a Heegaard splitting. \textit{Augmented Heegaard diagrams} have several desirable properties: each 2-cell is a square, they have \textit{non-positive combinatorial curvature} and they are \textit{virtually special}. Restricting to manifolds that do not have $S^1 \times S^2$ as a connect summand, augmented Heegaard diagrams are tied to the decomposition of a $3$-manifold via connect sum as described above.

math.GT↗
arXiv2022

Hyperbolic cone metrics and billiards

A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.

math.GT↗
arXiv2019

Structure on the Top Homology and Related Algorithms

We explore the special structure of the top-dimensional homology of any compact triangulable space $X$ of dimension $d$. Since there are no $(d+1)$-dimensional cells, the top homology equals the top cycles and is thus a free abelian group. There is no obvious basis, but we show that there is a canonical embedding of the top homology into a canonical free abelian group which has a natural basis up to signs. This embedding structure is an invariant of $X$ up to homeomorphism. This circumstance gives the top homology the structure of an (orientable) matroid, where cycles in the sense of matroids correspond to the cycles in the sense of homology. This adds a novel topological invariant to the topological literature. We apply this matroid structure on the top homology to give a polynomial-time algorithm for the construction of a basis of the top homology (over $\mathbb{Z}$ coefficients).

math.AT↗
arXiv2019

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.

math.GT↗