arXiv research

Survey of recent Kleinian representation convergence results.

This paper shows similarities in deformation spaces of Kleinian groups and anti-holomorphic maps.

Study on hyperconvex representations of hyperbolic groups in complex flag manifolds.

The paper proves rigidity for complex Kleinian groups.

New examples of embeddings defy Anosov representation limits.

Study of Eisenstein series linked to hyperbolic cusps.

Study of rank 2 Kleinian groups with specific parabolic elements.

In this paper, we give a complete criterion for a discrete faithful representation $ρ:F_n \ra \pslc$ to be primitive stable. This will answer Minsky's conjectures about geometric conditions on $\H^3/ρ(F_n)$ regarding the primitive stability of $ρ$.

New theorem proves rigidity of Kleinian group representations under specific conditions.

Cohomology fractals are visual representations of cohomology classes on hyperbolic 3-manifolds.

We discuss which Kleinian groups are commensurable with Kleinian groups generated by rotations, with particular emphasis on Kleinian groups that arise from Dehn surgery on a knot.

Revisiting a construction due to Vigneras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric.

Generalizes existence of bending laminations for Kleinian groups.

We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly det…

Agol's announcement proved a full classification of certain Kleinian groups.

Study counts and equidistributes tori in Kleinian group self-joinings.

Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.

The paper controls the geometry of surface subgroups in specific Kleinian groups.

The paper encourages Kleinian group thinking for higher rank Lie groups.

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π_1(M)) denote the space of (conjugacy classes of) discrete faithful representations of π_1(M) into PSL 2 (C). The components of the interior MP(π_1(M)) of AH(π_1(M)) (as a subset of the appropriate representation variety) ar…

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.

Constructs Kleinian groups from free groups via hyperbolization.

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…

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free t…

The paper classifies Kleinian groups with Hausdorff dimension less than 1.

In this paper we give a necessary and sufficient condition in which a sequence of Kleinian punctured torus groups converges. This result tells us that every exotically convergent sequence of Kleinian punctured torus groups is obtained by the method due to Anderson and Canary. Thus we obtain a complete description of th…

The study connects Kleinian group divergence to random walk recurrence.

Alternative proof classifies Kleinian groups with two parabolics.

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.

We prove that there are only finitely many arithmetic Kleinian maximal reflection groups.

We give a brief overview of the current state of the study of the deformation theory of Kleinian groups. The topics covered include the definition of the deformation space of a Kleinian group and of several important subspaces; a discussion of the parametrization by topological data of the components of the closure of …

One of the basic problems in studying topological structures of deformation spaces for Kleinian groups is to find a criterion to distinguish convergent sequences from divergent sequences. In this paper, we shall give a sufficient condition for sequences of Kleinian groups isomorphic to surface groups to diverge in the …

This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space for n greater than 3. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of the hyperbolic 3-space.

We present examples of hyperbolizable 3-manifolds $M$ with the following property. Let $CC(π_1(M))$ denote the space of convex co-compact representations of $π_1(M)$. We show that for every $K\geq 1$ there exists a representation $ρ$ in $\bar {CC(π_1(M))}$ so that every $K$-quasiconformal deformation of $ρ$ lies in the…

In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups $G_1$ and $G_2$ of an infinite co-volume Kleinian group $G \subset \Isom(\mathbf{H}^3)$ having $Λ(G_1) = Λ(G_2)$ are commensurable. In particular, it is proved tha…

We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.

We study orbital functions associated to Kleinian groups through the heat kernel approach developed in \cite{artmoiheatcounting1}.

We show that the space of Kleinian punctured torus groups is not locally connected.

We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a rou…

Overview of dynamics in algebraic correspondences and their connections.

We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35.

We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.

Study of groups acting on complex projective varieties.

Proves convergence groups on a 2-sphere are Kleinian groups.

We show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. This implies, in particular, that there exist only finitely many conjugacy classes of cocompact two generated arithmetic Kleinian groups. The proof of the main result is based on a generalized G…

In the paper `Automorphic functions for a Whitehead-complement group', [Osaka J Math 43 (2006) 63-77] Matsumoto, Nishi and Yoshida constructed automorphic functions on real 3-dimensional hyperbolic space for a Kleinian group called the Whitehead-link-complement group. For a Kleinian group (of the first kind), no automo…

New framework mated Kleinian groups with complex polynomials, revealing unique group properties.

Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.