Katsuhiko Matsuzaki

Inclusion relations between the Bers embeddings of Teichmüller spaces

We prove that if the Bers embeddings of the Teichmüller spaces of infinitely generated Fuchsian groups are coincident, then these Fuchsian groups are the same.

A countable Teichmüller modular group

We construct an example of a Riemann surface of infinite topological type for which the Teichmiiller modular group consists of only a countable number of elements. We also consider distinguished properties which the Teichmuller space of this Riemann surface possesses.

The Patterson–Sullivan measure and proper conjugation for Kleinian groups of divergence type

A Kleinian group (a discrete subgroup of conformal automorphisms of the unit ball) G is said to have proper conjugation if it contains the conjugate αGα−1 by some conformal automorphism α as a proper subgroup in it. We show that a Kleinian group of divergence type cannot have proper conjugation. Uniqueness of the Patterson–Sullivan measure for such a Kleinian group is crucial to our proof.

Conservative Action of Kleinian Groups with Respect to the Patterson-Sullivan Measure

For a non-elementary Kleinian group G acting on an n-dimensional sphere, we consider a conformally invariant probability measure of the dimension at the critical exponent of the Poincaré series. When this diverges, such a measure is unique and it is called the Patterson-Sullivan measure. We prove that the action of any non-trivial normal subgroup Г of G is conservative with respect to the Patterson-Sullivan measure for G.

A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space

For an analytically infinite Riemann surface R, the quasiconformal mapping class group MCG(R) always acts faithfully on the ordinary Teichmuller space T(R). However in this paper, an example of R is constructed for which MCG(R) acts trivially on its asymptotic Teichmuller space AT(R).

The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups

In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension.

Simply connected invariant domains of kleinian groups not in the closures of teichmüller spaces

In the first part of this note, we consider rigid domains which are invariant under certain elementary groups. Each of them corresponds to an isolated point of the set of Schwarzian derivatives of Schlicht functions in the Banach space of the quadratic differentials. Next, we consider “b-group”, that is, a Kleinian group which has a simply connected invariant component. For finitely generated Kleinian groups, all the b-groups are expected to be in the closures of the Bers embeddings of the Teichmuller spaces. We construct two kinds of counterexamples of this conjecture for infinitely generated Kleinian groups.

The action at infinity of conservative groups of hyperbolic motions need not have atoms

Herein the authors show that discrete groups of motions on H n +1 may be conservative on S n but have no positive measure ergodic components for this boundary action. An explicit example of such a group is given for H 3 using the Apollonian circle packing of R 2 .

Mackey and Frobenius structures on odd dimensional surgery obstruction groups

C. T. C. Wall formulated surgery-obstruction groups Ln(Z(G)) in terms of quadratic modules and automorphisms. C. B. Thomas showed that the Wall-group functors Ln(Z(−) ,w |−) are modules over the Hermitian-representation-ring functor G1(Z, −) if the orientation homomorph- ism w is trivial. A. Bak generalized the notion of quadratic module by introducing quadratic-form parameters, and obtained various K-groups related to quadratic modules and automorphisms. One of the authors established that some Bak groups Wn(Z(G) ,� ; w) are equivariant-surgery-obstruction groups and showed in the case of even dimension n that the Bak-group functor Wn(Z(−) ,� −; w|−) is a w-Mackey functor as well as a module over the Grothendieck-Witt-ring functor GW0(Z, −), where w is possibly nontrivial. In this paper, we prove the same facts in the case of odd dimension n. Mathematics Subject Classifications (2000): Primary 19G12, 19G24, 19J25; Secondary 57R67.

Dynamics of teichmüller modular groups and topology of moduli spaces of Riemann surfaces of infinite type

We investigate the dynamics of the Teichmüller modular group on the Teichmüller space of a Riemann surface of infinite topological type. Since the modular group does not necessarily act discontinuously, the quotient space cannot inherit a rich geometric structure from the Teichmüller space. However, we introduce the set of points where the action of the Teichmüller modular group is stable, and we prove that this region of stability is generic in the Teichmüller space. By taking the quotient and completion with respect to the Teichmüller distance, we obtain a geometric object that we regard as an appropriate moduli space of the quasiconformally equivalent complex structures admitted on a topologically infinite Riemann surface.