arXiv research

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.

Generalizes existence of bending laminations for Kleinian groups.

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.

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…

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

Survey of recent Kleinian representation convergence results.

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.

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.

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…

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

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

We give a full proof to Agol's announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.

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

Study measures invariant under horospherical subgroups for finitely generated Kleinian groups.

New groups found with critical exponents close to but less than max.

Survey connects hyperbolic groups to manifolds and Kleinian groups.

Uniform counting formulas for orthogeodesics in Kleinian groups converge.

We prove the existence of Cannon-Thurston maps for simply and doubly degenerate surface Kleinian groups. As a consequence we prove that connected limit sets of finitely generated Kleinian groups are locally connected.

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.

We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.

In this paper we prove that there exists a positive number $λ>0$, such that any 2-generated Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups.

In this paper we provide the complete classification of Kleinian groups of Hausdorff dimensions less than $1.$ In particular, we prove that every purely loxodromic Kleinian groups of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. As an application, the result of \cite{H} then implies…

Proof that specific groups are quasi-isometrically rigid.

The paper proves rigidity for complex Kleinian groups.

Combines Kleinian groups and polynomials into a dynamical system.

Study of Eisenstein series linked to hyperbolic cusps.

Using the rings of Lipschitz and Hurwitz integers $\mathbb{H}(\mathbb{Z})$ and $\mathbb{H}ur(\mathbb{Z})$ in the quaternion division algebra $\mathbb{H}$, we define several Kleinian discrete subgroups of $PSL(2,\mathbb{H})$

We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.