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Samuel A. Ballas

Publications and source records attributed to Samuel A. Ballas.

14 recordsLinked to original sources
arXiv2026

Geometric Perspective on Concentration Phenomena in Frame Theory

Parseval and equal-norm frames play a fundamental role in frame theory and signal processing. It is known that a random frame, with unit vectors drawn independently from the uniform distribution on the sphere, will be nearly Parseval with high probability; asymptotic results go back at least to Goyal, Vetterli, and Thao and a non-asymptotic error bound was proved more recently by Kwok, Lau, and Ramachandran. In this work, we prove a dual result, which shows that random Parseval frames, with respect to the Haar measure, are nearly equal-norm with high probability. Our proofs are geometric in nature, and rely on general measure concentration principles in Riemannian manifolds. Using these techniques, we also give a novel probabilistic upper bound for the Paulsen problem.

math.FA↗
arXiv2026

Parabolic-preserving deformations of cusped hyperbolic lattices

We study deformations of non-cocompact lattices of ${\rm SO}(n,1)$ into ${\rm SU}(n,1)$ and ${\rm SO}(n+1,1)$. A necessary condition for these deformations to remain discrete and faithful (when $n \geqslant 3$) is for the parabolic subgroups to remain parabolic and discrete; we call such representations \emph{strongly parabolic-preserving}. We show that the figure-eight knot group admits a one-parameter family of Zariski-dense parabolic-preserving deformations into ${\rm SU}(3,1)$, with further deformations into ${\rm SU}(2,2)$. We also study the \emph{bending deformations} of the Bianchi groups (seen as subgroups of ${\rm SO}(3,1)$) along the modular surface into ${\rm SU}(3,1)$ and ${\rm SO}(4,1)$, and show that infinitely many of them are strongly parabolic-preserving in ${\rm SU}(3,1)$, while none are strongly parabolic-preserving in ${\rm SO}(4,1)$. Finally, for any $n \geqslant 3$, we show that there exist infinitely many non-commensurable cusped hyperbolic $n$-manifolds whose corresponding hyperbolic representation admits a 1-parameter family of parabolic-preserving deformations into ${\rm SU}(n,1)$.

math.GT↗
arXiv2023

On the existence of Parseval frames for vector bundles

Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles -- this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors -- and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of $G$-bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.

math.DG↗
arXiv2023

Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

Suppose a relatively elliptic representation $ρ$ of the fundamental group of the thrice-punctured sphere $S$ is given. We prove that all projective structures on $S$ with holonomy $ρ$ and satisfying a tameness condition at the punctures can be obtained by grafting certain circular triangles. The specific collection of triangles is determined by a natural framing of $ρ$. In the process, we show that (on a general surface $Σ$ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their Möbius completion, and in terms of certain meromorphic quadratic differentials.

math.GT↗
arXiv2020

Thin subgroups isomorphic to Gromov--Piatetski-Shapiro lattices

In this paper we produce many examples of thin subgroups of special linear groups that are isomorphic to the fundamental groups of non-arithmetic hyperbolic manifolds. Specifically, we show that the non-arithmetic lattices in $\mathrm{SO}(n,1)$ constructed by Gromov and Piatetski-Shapiro can be embedded into $\mathrm{SL}_{n+1}(\mathbb{R})$ so that their images are thin subgroups

math.GT↗
arXiv2020

The Moduli Space of Marked Generalized Cusps in Real Projective Manifolds

In this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to $M \times [0, \infty)$ where $M$ is a closed Euclidean manifold. These are classified in [2]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of $π_1(M)$. It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on $M$, and the fiber is a closed cone in the space of cubic differentials. For 3-dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.

math.GT↗
arXiv2020

Properly convex bending of hyperbolic manifolds

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $Σ$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geq 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds. Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.

math.GT↗
arXiv2019

Gluing equations for real projective structures on 3-manifolds

Given an orientable ideally triangulated $3$--manifold $M$, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on $M$. These equations represent a unifying framework for the classical Thurston gluing equations in hyperbolic geometry and their more recent counterparts in Anti-de Sitter and half-pipe geometry. Moreover, these equations can be used to detect properly convex structures on $M$. The paper also includes a few explicit examples where the equations are used to construct properly convex structures.

math.GT↗
arXiv2019

Generalized Cusps in Real Projective Manifolds: Classification

A generalized cusp $C$ is diffeomorphic to $[0,\infty)$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $Γ$, in one of a family of affine groups $G(ψ)$, parameterized by a point $ψ$ in the (dual closed) Weyl chamber for $SL(n+1,\mathbb{R})$, and $Γ$ determines the cusp up to equivalence. These affine groups correspond to certain fibered geometries, each of which is a bundle over an open simplex with fiber a horoball in hyperbolic space, and the lattices are classified by certain Bieberbach groups plus some auxiliary data. The cusp has finite Busemann measure if and only if $G(ψ)$ contains unipotent elements. There is a natural underlying Euclidean structure on $C$ unrelated to the Hilbert metric.

math.GT↗
arXiv2019

Rank 1 deformations of non-cocompact hyperbolic lattices

Let $X$ be a negatively curved symmetric space and $Γ$ a non-cocompact lattice in $\rm{Isom}(X)$. We show that small, parabolic-preserving deformations of $Γ$ into the isometry group of any negatively curved symmetric space containing $X$ remain discrete and faithful (the cocompact case is due to Guichard). This applies in particular to a version of Johnson-Millson bending deformations, providing for all $n$ infnitely many non-cocompact lattices in ${\rm SO}(n,1)$ which admit discrete and faithful deformations into ${\rm SU}(n,1)$. We also produce deformations of the figure-8 knot group into $\rm{SU}(3,1)$, not of bending type, to which the result applies.

math.GT↗
arXiv2017

Convex projective structures on non-hyperbolic three-manifolds

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist's theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.

math.GT↗
arXiv2015

Finite Volume Properly Convex Deformations of the Figure-eight Knot

In this paper we show that some open set of the representations of the fundamental group of figure-eight knot complement found in \cite{Ballas12a} are the holonomies of a family of finite volume properly convex projective structures on the figure-eight knot complement.

math.GT↗
arXiv2014

Deformations of Non-Compact, Projective Manifolds

In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-trivial, strictly convex deformations near their complete hyperbolic structure.

math.GT↗