In 2004-2005, the seminar was organised by Javier Aramayona and Akira Ushijima.

James Giblin (Warwick)

*Sullivan manifolds, Stable Homeomorphisms and Simplicity*

**Abstract:**
A Sullivan manifold is a closed hyperbolic n-manifold for which the
complement of any point can be immersed
into R^{n}. In 1977 Sullivan proved the existence of such manifolds in
every dimension. This allowed techniques
which had been previously successful in proving topological theorems to be
applied to the analogous problems in
the quasiconformal and bilipschitz categories. I will explain how and
present another result in this vain.
Namely, that the groups of quasiconformal and bilipschitz homeomorphisms
of S^{n} are simple.

Vivien Easson (Oxford)

*Handlebody surgery of 3-manifolds via the knotted Y graph*

**Abstract:**
I will discuss a new area of interest in the study of 3-manifolds:
handlebody surgery.
This is a generalization of Dehn surgery, where we cut out a neighbourhood
of a knotted curve
and glue in a solid torus to create a new 3-manifold. In handlebody
surgery the curve is replaced
by a graph and the solid torus by a handlebody of higher genus. I will
start by giving an introduction
to Dehn surgery, which has been extensively studied over the last 30
years, and then explain some of the
difficulties involved in generalizing to higher genus. Throughout I will
illustrate the ideas involved by
looking at particular examples.

Daan Krammer (Warwick)

*Still more lattices for braid groups*

**Abstract:**
Two well-known left-invariant lattice orderings on the braid group are
the Garside one (1969) and the Birman-Ko-Lee
on (1998). These are also called Garside structures. A mild
generalisation of this concept is obtained by looking at groupoids
rather than groups.
After presenting some preliminaries, I will present a new class of
groupoid-Garside-structures on pure braid groups, one for each function
from the set of strands to the integers >1. For constant functions
one gets groupoid-Garside-structures on braid groups.
These results were inspired by the question to find Garside structures
on surface mapping class groups. The above construction easily
generalises to higher genus. The result is an ordered set on which the
mapping class group acts, but
the ordered set is not a lattice unfortunately.

Jim Anderson (Southampton)

*Conformal measures associated to ends of hyperbolic manifolds*

Bas Lemmens (Warwick)

*Hilbert's geometry and dynamics of cone maps*

John Parker (Durham)

*Euclidean cone metrics on the sphere and lattices in complex
hyperbolic space*

Ser Tan (Singapore)

*Generalized Markoff maps and McShane's identity*

**Abstract:**We study the character variety of representations of
the free group on two generators into SL(2,C) via generalised
Markoff maps, following B. Bowditch. Of central interest are the
representations satisfying some simple finiteness
conditions called the Bowditch Q-conditions. Applications include
generalizations of McShane's identity, dynamics of the mapping class
group action on the relative character varieties, and identities for
closed hyperbolic 3-manifolds obtained by hyperbolic Dehn-surgery on
once-punctured torus bundles over the circle.

Ruth Kellerhals (Fribourg)

*Polylogarithms, Hyperbolic Volume and Mahler measure*

**Abstract:**
We present a survey about hyperbolic volume computations in terms
of polylogarithms. We discuss new developments in dimension three and
report about connexions with respect
to Mahler measures

Brian Bowditch (Southampton)

*Hyperbolic 3-manifolds and the geometry of the curve complex*

*There will be no seminar this week*

Brent Everitt (York)

*Coxeter groups and hyperbolic manifolds*

Giannis Platis (Durham)

*Complex hyperbolic quasifuchsian groups*

Adam Epstein (Warwick)

*Simultaneous Uniformization and Limits, for Kleinian Groups and
Rational Maps*

**Abstract:**

Bers proved in 1960 that any pair of homeomorphic compact Riemann surfaces may be simulatenously uniformized by a quasifuchsian group. This observation shows that the space of (Mobius conjugacy classes of) such groups has a canonical product structure: the factors being the relevant Teichmuller space and its complex conjugate. Around 1980, Kerckhoff-Thurston proved that when the genus is greater than 1 (equivalently, when the Teichmuller space has dimension greater than 1) this product DOES NOT extend continuously to the closure in the space of all groups.

There is a parallel phenomenon in holomorphic dynamics. For each D>1, the natural deformation space of the rational map z -> z^D splits as a product of Teichmuller(-like) spaces. McMullen conjectured that when D>2 the corresponding product structure should have discontinuities as the boundary. We have proved a stronger version of this conjecture.

Despite the formal similarity of these results, the actual details involved are extremely context dependent. For Kleinian groups one examines the quotient 3-manifold, something which is unavailable in holomorphic dynamics. Conversely, in holomorphic dynamics one examines the invariants of parabolic points - these invariants being trivial for Mobius transformations.

Ian Short (Cambridge)

*Continued Fractions, M\"obius Transformations and Hyperbolic
Geometry*

**Abstract:**
The Stern-Stolz Theorem states that if the infinite series $\sum|b_n|$
converges then the continued fraction $\mathbf{K}(1|b_n)$
diverges. In 1948, H.S.Wall posed the question as to whether just
convergence, rather than absolute convergence, of $\sum b_n$
is sufficient for divergence of $\mathbf{K}(1|b_n)$. We investigate the
relationship between $\sum|b_n|$ and $\mathbf{K}(1|b_n)$ with
hyperbolic geometry and use this geometry to construct a sequence $b_n$
of real numbers for which both $\sum b_n$ and $\mathbf{K}(1|b_n)$
converge, thereby
answering Wall's question.

Mark Pollicott (Porto and Warwick)

*Some applications of dynamical systems to problems in hyperbolic
geometry*

**Abstract:**
I will give a fairly relaxed survey of some applications of ideas from
dynamical systems to certain problems arising in
hyperbolic dynamics. In particular, I will discuss methods of computing
the dimension of limit sets for Kleinian groups
(older work with Oliver Jenkinson) and pair correlation results for
lengths of closed geodesics on hyperbolic surfaces (recent
work with Richard Sharp).

Kirill Krasnov

*Towards the quantum geometry of hyperbolic 3-manifolds*

**Abstract:**In broad terms, the programme of quantum geometry is
applying the usual rules of quantum nechanics to geometry or geometric
structure.
In this respect the programme is similar to non-commutative geometry,
where the space of functions on some space gets deformed and becomes
non-commutative.
Unlike non-commutative geometry, quantum geometry applies quantum
mechanics not to the space of functions, but to the geometry of the
space itself. I will
explain how this programme is most successful in 3 dimensions, where it
leads to familiar topological invariants of 3-manifolds. The known
invariants can be
interpreted as coming from quantization of the flat, and positively
curved 3-dimensional geometric structures. I will then review the
status of quantization of the negatively curved,
or hyperbolic 3-dimensional geometry. An outcome of this quantization
programme, when completed, will be new topological invariants of
3-manifolds.

Norbert Peyerimhoff (Durham)

*Isoperimetric and extremal properties of regular polyhedra in
hyperbolic space*

**Abstract:**
The classical isoperimetric problem asks for the body of minimal
surface amongst all convex bodies of fixed volume.
There are many ways to modify this extremality problem: one can
consider extremalities of other characteristic geometric values
(instead of surface area), one can consider smaller sets of convex
bodies (e.g. particular sets of polyhedra), and one can also change the
underlying geometry to be a space of constant curvature.
In this talk we discuss particular extremality problems of polyhedra in
the n-dimensional real hyperbolic space. The results
are based on a hyperbolic version of the Steiner
symmetrization.

Cyril Lecuire (Warwick)

*Doubly incompressible measured laminations are binding*

**Abstract:**
An important step in Thurston's hyperbolization of fibered 3-manifolds is
the Double Limit Theorem.
Let us consider the set of representations of a surface group in PSL(2,C)
for which the length of a
given pair of binding measured laminations is bounded by some given
constant. The Double Limit Theorem states that,
modulo conjugacy, this set is compact. I will explain how to extend this
result to hyperbolic manifolds with
compressible boundary. In this case the pair of binding laminations is
replaced by a doubly incompressible lamination
on the boundary of the manifold. This is joint work with I. Kim and K.
Oshika.

Alastair Fletcher (Warwick)

*Local rigidity of infinite dimensional Teichmüller spaces*

**Abstract**:
In this talk I will show how, for any two hyperbolic Riemann surfaces
which are not of finite analytic type, their Bergman spaces
are isomorphic and, as a consequence of this, how their Teichmüller
spaces are locally bi-Lipschitz equivalent.

Dan Goodman (Warwick)

*Spirals in slices of quasifuchsian space*

**Abstract**:
In this talk I will prove the folklore/conjecture that the Maskit and Bers
slices of the quasifuchsian space
of a once-punctured torus spiral to an indefinite (but not infinite)
extent near every point on their boundaries.

*****Cancelled*****

Ken Shackleton (Southampton)

*Computing distances in the curve complex*

**Abstract**:
We give explicit bounds on the intersection number between any curve on
a tight geodesic and the two ending curves. We use this
to construct at least one geodesic between any two vertices, to
construct all tight geodesics and to conclude that distances are
computable. Our algorithm
applies to all surfaces. The central argument makes no use of the
geometric limit arguments seen in the recent work of Bowditch (2003)
and Masur-Minsky (2000).
From this we recover the finiteness result of Masur-Minsky for tight
geodesics.

Javier Aramayona (Warwick)

*The pants complex and mapping class groups*

**Abstract**:
In this talk, we will construct a compactification for a fine,
hyperbolic graph, as a generalisation of a construction by Bowditch. We
will discuss possible applications of this to the pants complex and the
mapping class group of the five-holed sphere.

Akira Ushijima (Warwick, Kanazawa University)

*Decorated Teichmüller space and its cell
decomposition*

**Abstract**:
The purpose of this talk is to explain the idea of the cell decomposition
of the decorated Teichmueller space of punctured surfaces by
R. C. Penner, following his paper "The decorated
Teichmüller space of punctured surfaces" in
Comm. Math. Phys. 113 (1987), 299-339. Here the
"decoration" means a positive weight on each cusp. A key tool
for obtaining this decomposition is the so-called "convex hull
construction" of a cell decomposition of a hyperbolic manifold, which
will also be explained in the talk. Some related topics will be
mentioned, if time allows.