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Pallavi Panda

Publications and source records attributed to Pallavi Panda.

6 recordsLinked to original sources
arXiv2025

Flip-graphs of non-orientable filling surfaces

Consider a surface $\Sigma$ with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface $\Sigma_n$ by singling out one of the boundary components and denoting by $n$ the number of marked points it contains. We consider the triangulations of $\Sigma_n$ whose vertices are the marked points and the associated flip-graph $\mathcal{F}(\Sigma_n)$. Quotienting $\mathcal{F}(\Sigma_n)$ by the homeomorphisms of $\Sigma$ that fix the privileged boundary component results in a finite graph $\mathcal{MF}(\Sigma_n)$. Bounds on the diameter of $\mathcal{MF}(\Sigma_n)$ are available when $\Sigma$ is orientable and we provide corresponding bounds when $\Sigma$ is non-orientable. We show that the diameter of this graph grows at least like $5n/2$ and at most like $4n$ as $n$ goes to infinity. If $\Sigma$ is an unpunctured M\"obius strip, $\mathcal{MF}(\Sigma_n)$ coincides with $\mathcal{F}(\Sigma_n)$ and we prove that the diameter of this graph grows exactly like $5n/2$ as $n$ goes to infinity.

math.GT↗
arXiv2025

Polyhedral realisations of finite arc complexes using strip deformations

We study infinitesimal deformations of complete hyperbolic surfaces with boundary and with ideal vertices, possibly decorated with horoballs. ``Admissible'' deformations are the ones that pull all horoballs apart; they form a convex cone of deformations. We describe this cone in terms of the arc complex of the surface: specifically, this paper focuses on the surfaces for which that complex is finite. Those surfaces form four families: (ideal) polygons, once-punctured polygons, one-holed polygons (or ``crowns''), and M\"obius strips with spikes. In each case, we describe a natural simplicial decomposition of the projectivised admissible cone and of each of its faces, realizing them as appropriate arc complexes.

math.DG↗
arXiv2023

Parametrisation of decorated Margulis spacetimes using strip deformations

Margulis spacetimes are complete affine 3-manifolds that were introduced to show that the cocompactness condition of Auslander's conjecture is necessary. There are Lorentzian manifolds that are obtained as a quotient of the three dimensional Minkowski space by a non-abelian free group acting properly discontinuously by affine isometries. Goldman-Labourie-Margulis showed that such a group is determined by a complete hyperbolic metric on a possibly non-orientable finite-type hyperbolic surface together with an infinitesimal deformation of this metric that uniformly lengthens all non-trivial closed curves on the surface. Furthermore, the set of all such infinitesimal deformations forms an open convex cone. Danciger Gu\'eritaud-Kassel parametrised the moduli space of Margulis spacetimes, with a fixed convex cocompact linear part, using the pruned arc complex. The parametrisation is done by gluing infinitesimal hyperbolic strips along a family of embedded, pairwise disjoint arcs of the hyperbolic surface that decompose it into topological disks. We generalise this result to complete finite-area hyperbolic surfaces with spikes decorated with horoballs. These are closely related to Margulis spacetimes decorated with finitely many pairwise disjoint affine light-like lines, called photons.

math.GT↗
arXiv2023

The arc complexes of partially decorated hyperbolic polygons

We consider two families of hyperbolic polygons: ideal and ideal once-punctured, some of whose spikes are decorated with horoballs. We show that the arc complexes of these two families of surfaces, generated by edge-to-edge arcs and edge-to-decorated-spike arcs, are closed piecewise linear balls. This is proved in a completely combinatorial setting: compact polygons whose vertices are assigned red or blue colouring. In order to prove the ballness, we show that these simplicial complexes are pseudo-manifolds and use shellability to conclude. As a consequence, we parametrise weakly-lengthening deformations of the partially decorated hyperbolic polygons.

math.CO↗
arXiv2023

Strip deformations of decorated hyperbolic polygons

In this paper we study the hyperbolic and parabolic strip deformations of ideal (possibly once-punctured) hyperbolic polygons whose vertices are decorated with horoballs. We prove that the interiors of their arc complexes parametrise the open convex set of all uniformly lengthening infinitesimal deformations of the decorated hyperbolic metrics on these surfaces, motivated by the work of Danciger-Gu\'eritaud-Kassel.

math.GT↗