# Hyperbolic Geometry Seminar 2004-2005

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

### Thursday 19th May

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.

### Thursday 12th May

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.

### Thursday 5th May

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.

### Thursday 28th April

Jim Anderson (Southampton)

Conformal measures associated to ends of hyperbolic manifolds

### Thursday 21st April

Bas Lemmens (Warwick)

Hilbert's geometry and dynamics of cone maps

### Thursday 10th March, 17:00, Room B1.01

John Parker (Durham)

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

### Thursday 10th March, 13:30, Room B3.02

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.

### Friday 25th February, 11am

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

### Thursday 17th February

Brian Bowditch (Southampton)

Hyperbolic 3-manifolds and the geometry of the curve complex

### Thursday 10th February

There will be no seminar this week

### Thursday 3rd February

Brent Everitt (York)

Coxeter groups and hyperbolic manifolds

### Thursday 27th January

Giannis Platis (Durham)

Complex hyperbolic quasifuchsian groups

### Thursday 20th January

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.

### Thursday 13th January

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.

### Thursday 2nd December

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).

### ****Cancelled/Postponed****

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.

### 25th November

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.

### 18th November

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.

### 11th November

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.

### 4th November

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.

### 28th October

*****Cancelled*****

### 21st October

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.

### 14th October

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.

### 7th October

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.

Current year of the seminar.