arXiv subjects

Ken'ichi Ohshika

Publications and source records attributed to Ken'ichi Ohshika.

23 records · Page 2Linked to original sources

arXiv2011

Reduced Bers boundaries of Teichmüller spaces

We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space into a point. This reduced Bers boundary turns out to be independent of the basepoint, and the action of the mapping class group on the Teichmüller space extends continuously to this boundary. We show that every auto-homeomorphism on the reduced Bers boundary comes from an extended mapping class. We also give a way to determine the limit in the reduced Bers boundary up to some ambiguity for parabolic curves for a given sequence in the Teichmüller space, by generalising the Thurston compactification.

math.GT↗

arXiv2010

Realising end invariants by limits of minimally parabolic, geometrically finite groups

We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and end invariants. This shows that the Bers-Sullivan-Thurston density conjecture follows from Marden's conjecture proved by Agol, Calegari-Gabai combined with Thurston's uniformisation theorem and the ending lamination conjecture proved by Minsky, partially collaborating with Masur, Brock and Canary.

math.GT↗

arXiv2008

Constructing geometrically infinite groups on boundaries of deformation spaces

Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup Ω_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit $Γ$ of quasi-conformal deformations of $G$ such that there is a homeomorphism $h$ from $\mathrm{Int} M$ to $\H^3/Γ$ compatible with the natural isomorphism from $G$ to $Γ$, the given laminations are unrealisable in $\H^3/Γ$, and the given conformal structures are pushed forward by $h$ to those of $\H^3/Γ$. Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden's conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur.

math.GT↗

arXiv2006

On Klein-Maskit Combination Theorem in space I

In this paper, we generalise the first Klein-Maskit combination theorem to discrete groups of Möbius transformations in higher dimensions. As a simple application of the main theorem, some examples will be constructed.

math.CV↗

arXiv1998

Divergent sequences of Kleinian groups

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 deformation spaces.

math.GT↗