arXiv research

Estimates small eigenvalues for geometrically finite manifolds.

Study essential spectrum of differential operators on geometrically finite orbifolds.

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…

Lower bounds for Hodge-Laplacian spectrum on orbifolds.

The paper characterizes geometrically finite surfaces via geodesic covers.

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically fini…

The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.

Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.

Study equidistribution for flows on geometrically finite convergence group actions.

Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.

Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of the Euclidean three dimensional space. In a recent preprint J. Jorge and F. Merc…

We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previous…

We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume…

The study classifies geometrically finite polynomials on the boundary of Blaschke products.

The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.

We develop a geometric scattering theory for a geometrically finite group acting on (a vector bundle over) a symmetric space of negative curvature. In particular, we obtain the meromorphic continuation of Eisenstein series and scattering matrices and their functional equations.

The paper proves geometrical finiteness for automorphism groups of K3 surfaces and related varieties.

Proves Singer conjecture for specific geometric varieties.

We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for manifolds commensurable with these groups, immersed incompressible surfaces lift to embe…

In this paper, we proved that any closed orientable 3-manifold 1-dominates at most finitely many geometric 3-manifolds.

Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.

The paper characterizes arithmetic metrics in coarsely geometric settings.

We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.

The paper shows geometric realisation over specific groups and knots.

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompr…

Constructs knots from 3-manifolds with specified geometric limits.

This is a summary of the material for 3 lectures on geometrically finite and infinite Kleinian groups delivered by the author at a workshop held at Tata Institute of Fundamental Research in April 2014.

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compres…

We show that minimal length carrier graphs are not unique, but if M is in a large class of hyperbolic 3-manifolds, including the geometrically finite ones, then M has only finitely many minimal length carrier graphs and no two of them are homotopic. As a corollary, we obtain a new proof that the isometry group of a geo…

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduc…

Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then $M^n$ is homeomorphic to the sphere $S^n$. Also, we define a concept of finite geometrical type and prove that finite geometrical type h…

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh.…

We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with geometrically finite limit. This allows us to show that the renormalized volume attains its…

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-ma…

In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper…

Researchers address the generation of differential invariants for geometric structures.

In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.

Proves symmetric spaces for certain quasi-isometric properties.

Researchers find a way to bound the complexity of certain subgroup geometric invariants.

Extends geometric decompositions to arbitrary meshes and forms.

Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was prove…

Extensions of Veech groups using hierarchical hyperbolic spaces.

The study explores conformal planes with finite areas.

Finite-time queue peaks in stochastic networks have logarithmic scaling after geometric thresholds.

New insights into Anosov representations of hyperbolic groups.

Study bottom of spectra on orbifolds via coverings.

New examples of subgroups in mapping class groups are found.