Complex Analysis and Dynamics Seminar

Fall 2007 Schedule

Sept. 7: Yunping Jiang (Queens College and GC of CUNY)
Bounded Geometry and Thurston Type Theorems for Branched Coverings

Thurston's theorem for critically finite branched coverings represents an important development in complex dynamical systems. It shows that a critically finite rational map is rigid and gives a necessary and sufficient condition for a critically finite branched covering to be realizable by a rational map. In this talk, I will review the theorem and discuss some new Thurston type results for geometrically finite and rotation type branched coverings, all from bounded geometry point of view.

Sept. 28: Perry Susskind (Connecticut College)
The Margulis Region and Continued Fractions

A hyperbolic manifold M may be understood as the quotient of hyperbolic space H by a torsion-free discrete group G of isometries of H. In dimensions 2 and 3 (i.e., when G is a Fuchsian or Kleinian group acting on hyperbolic 2- or 3-space), the "thin" pieces of M are understood largely by noting that for each parabolic fixed point p of G, there is a set with a quite simple shape called a horoball region that is invariant under the stabilizer of p in G. Sadly(?), in dimensions greater than 3 there need not be an invariant horoball region, but there is always an invariant Margulis region whose more complicated shape depends only upon the stabilizer of p in G. An explicit description of the shape of the Margulis region is given for a parabolic fixed point p with a rank one stabilizer in a discrete group of isometries acting on hyperbolic 4-space. The particular shape of this region depends on the continued fraction expansion of the rotational part of the generator of the stabilizer of p.

Oct. 5: Dragomir Saric (Queens College of CUNY)
The Teichmuller Distance Between Finite Index Subgroups of PSL(2,Z)

For a given t > 0, we show that there exist two finite index subgroups of PSL(2,Z) which are (1+t)-quasisymmetrically conjugate but the conjugating homeomorphism is not conformal. This implies that for any t > 0 there are two finite regular covers of the once-punctured modular torus T0 (or just the modular torus) and a (1+t)-quasiconformal mapping between them that is not homotopic to a conformal map. As an application of the above results, we show that the closure of the orbit of the basepoint in the Teichmuller space T(S) of the punctured solenoid S under the action of the corresponding modular group is strictly larger than the orbit and is necessarily uncountable. This is joint work with V. Markovic.

Oct. 12: Youngju Kim (Graduate Center of CUNY)
On Isometries of Hyperbolic 4-Space

It has been known that orientation-preserving isometries of hyperbolic (n+2)-space have 2x2 matrix representations using the Clifford algebra Cn of n generators. Recall that
Isom(H2)= PSL(2,R),
Isom(H3)= PSL(2,C), and
Isom(H4)= PSL(2,Q),
where we identify real numbers R with C0, complex numbers C with C1, and quaternions Q with C2. These isomorphisms suggest a similar approach to the study of hyperbolic isometries in higher dimensions. We will carry out such a study, emphasizing the 4-dimensional case.

Oct. 19: Ren Guo (Rutgers University)
Parameterizations of the Teichmuller Space of Surfaces with Boundary

For a fixed ideal triangulation, the edge lengths parameterize the Teichmuller space of surfaces with boundary. F. Luo found a family of new coordinates of the Teichmuller space from the derivative cosine law. Under each of the new coordinates, the Teichmuller space is an open convex polytope. These new coordinates can be considered as an analog of R. Penner's simplicial coordinate of the decorated Teichmuller space of surfaces with cusps.

Oct. 26: Youngju Kim (Graduate Center of CUNY)
On Marden Stability in Hyperbolic 4-Space

This is a sequel to my October 12 talk. An n-dimensional Mobius group is said to be quasiconformally stable if its sufficiently small deformations in Isom+(Hn) are all quasiconformally conjugate to it. For example, in dimension 3, a Kleinian group corresponding to the trice-punctured sphere is quasiconformally stable. More generally, Marden has shown that any geometrically finite Kleinian group in dimension 3 must be quasiconformally stable. Here we present an example of a trice-punctured sphere group in dimension 4 which is geometrically finite but not quasiconformally stable.

Nov. 30: Bernard Maskit (Stony Brook University)
Some One-Dimensional Deformation Spaces

A metric on a compact oriented 4-manifold with no boundary is optimal if it minimizes the square norm of the curvature tensor, while keeping the total volume fixed. In a series of papers, LeBrun has studied the question of which simply connected 4-manifolds admit optimal metrics that are not Einstein (every Einstein metric on such a manifold is optimal). One such construction leads to the following question: "Does there exist a Kleinian group of the second kind that is a combination theorem free product of a cyclic group of order 2 and a cyclic group of order 3, where the Hausdorff dimension of the limit set of this group is greater than 1?" We discuss the deformation spaces of Kleinian groups that are combination theorem free products of two elliptic cyclic groups, including an answer to the above question. This is joint work with Claude LeBrun.

Dec. 7: Caroline Series (University of Warwick)
Excursions into the Thin Part of Teichmuller Space

We discuss Rafi's combinatorial conditions which allow one to detect whether a curve gets short in some surface along a Teichmuller geodesic. We explain how these conditions can be used to compare Teichmuller geodesics to "lines of minima." Introduced by Kerkchoff, these are also bi-infinite paths in the Teichmuller space determined by the same data, namely a pair of measured laminations which fill up the surface. We get an estimate of the distance between the two paths, leading to the result that lines of minima are Teichmuller quasi-geodesics. This is joint work with Young Eun Choi and Kasra Rafi.

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