arXiv research

New findings on cusped Borel Anosov representations and their properties.

Study pressure metrics for cusped Hitchin representations.

Characterizes holonomies of convex projective cusps.

Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

Study of Eisenstein series linked to hyperbolic cusps.

Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.

Entropy rigidity theorem for cusped Hitchin representations.

We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We…

Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.

The paper classifies a space of generalized cusps and its moduli.

Develops correlation number for specific potentials and Hitchin representations.

We construct a representation of the braid groups in a cluster C*-algebra coming from a triangulation of the Riemann surface S with one or two cusps. It is shown that the Laurent polynomials attached to the K-theory of such an algebra are topological invariants of the closure of braids. In particular, the Jones and HOM…

Let $M$ be a cusped hyperbolic $3$-manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in $\mathrm {PGL}(2,\mathbf {C})$ (up to conjugation) is of complex dimension the number $ν$ of cusps near the hyperbolic representation. It seems natural to ask whether some …

It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms which preserve some bi-ordering of the free group rank n. As a consequence of our w…

Classifies worst approximable rational numbers using hyperbolic geometry.

A planar portrait of a manifold is the pair of the image and the critical values of the manifold through a stable map into the plane. It can be considerd a geometric representation of the manifold drawn in the plane. The cusped fan is its basic local configuration. In this article, we focus on the fibreing structure ov…

We show that Norbury's McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of pi_1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.

We survey some of our recent results on length series identities for hyperbolic (cone) surfaces, possibly with cusps and/or boundary geodesics; classical Schottky groups; representations/characters of the one-holed torus group to $SL(2, \mathbf C)$; and hyperbolic 3 manifolds obtained by hyperbolic Dehn surgery on punc…

New divide with gleams method simplifies symmetric link representation.

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL_2(C) to the corresponding Reidemeister torsion. Our proof rests on an expression of the analytic torsion in terms of …

Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.

Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety around this representation. As a corollary, this representation is isolated among al…

Let W be a compact manifold and let ρbe a representation of its fundamental group into PSL(2,C). The volume of ρis defined by taking any ρ-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume doe…

The Ptolemy coordinates for boundary-unipotent SL(n,C)-representations of a 3-manifold group were introduced in Garoufalidis-Thurston-Zickert inspired by the A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we define the Ptolemy field of a (generic) PSL(2,\C)-representation and prove …

4-manifolds show every flat 3-manifold as cusp sections.

Study of parabolic-preserving deformations of hyperbolic lattices.

It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and…

Study geodesics entering a fixed cusp neighborhood multiple times.

Characterizes monodromies of projective structures on finite-type surfaces.

We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing d…

The paper classifies different types of cusps on plane curves.

The paper characterizes links in 3D from divides with cusps.

The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.

Proves EGF representations in specific geometric contexts.

Let $G$ be a real simple linear connected Lie group of real rank one. Then, $X := G/K$ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic hyperbolic space or the Cayley hyperbolic plane. We define the Schwartz space $…

The paper extends and applies a new shrinkage prior in Bayesian factor analysis.

Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.

Study of rank 2 Kleinian groups with specific parabolic elements.

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling w…

Four hyperbolic 24-cell 4-manifolds with one cusp are identified.

This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orb…

We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume $v_m = 4π^2/3$ and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has lowest volume among…

Study Dirac operator on cusped hyperbolic manifolds, finding spectrum properties.

Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.

Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.

Recent work of Ballas, Cooper, and Leitner identifies $(n+1)$ types of $n$-dimensional convex projective cusps, one of which is the standard hyperbolic cusp. Work of Ballas-Marquis, and Ballas-Danciger-Lee give examples of these exotic (non-hyperbolic) type cusps in dimension $3$. Here an extension of the techniques of…

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…