# Geometry Seminar 2005-2006

In 2005-2006, the seminar was organised by Javier Aramayona and Samuel Lelièvre.

### Thursday 29th September (in Room B1.01)

Cliff Earle (Cornell University)

The genus two Jacobians that are isomorphic to a product of elliptic curves.

Abstract: Every compact Riemann surface $R$ has a Jacobi variety (often called its Jacobian) $J(R)$, which is a complex torus with some extra structure that makes it a principally polarized Abelian variety and allows us to reconstruct $R$ from $J(R)$. I shall describe this briefly from scratch. The dimension of $J(R)$ equals the genus of $R$. When the genus is two, we can ask whether $J(R)$ is (isomorphic to) the product of two one-dimensional tori (elliptic curves). The answer is no if the isomorphism is required to respect the polarization. But many examples are known where $J(R)$ is isomorphic as a complex manifold to a product of two elliptic curves. This talk will list all possible examples. The proof that our list is complete is surprisingly elementary.

### Thursday 6th October

Kirill Krasnov (Nottingham)

Equidistant foliations and volumes of co-compact hyperbolic 3-manifolds.

Abstract: We review some well-known results about foliations of constant curvature 3-manifolds M by surfaces equidistant to a given one. Given the first and second fundamental forms of a surface S, the 3-manifold metric can be written down explicitly in the Gaussian coordinate system based on S, at least in the neighbourhood of S. We apply this description to convex co-compact hyperbolic 3-manifolds M and consider two cases: (i) S is a minimal surface; (ii) S is a convex surface. In the first case we are led to the notion of almost-fuchsian manifolds which contain a unique minimal surface S and are foliated by surfaces equidistant to S. We also show that not all quasi-fuchsian manifolds are almost-fuchsian. In the second case we get an equidistant foliation of a hyperbolic end of M. Given such a foliation for each hyperbolic end one can define the so-called renormalized volume of M. We compute the renormalized volume explicitly and show that it is related to certain other volume. We show that this other volume is given by the so-called Liouville action functional for the so-called horospherical metric built from the first and second fundamental forms of S.

### Thursday 13th October

Samuel Lelièvre (Warwick)

Square-tiled surfaces and moduli spaces of abelian differentials

Abstract: We will introduce square-tiled surfaces and discuss how these combinatorial objects can give insight on the moduli spaces of abelian differentials in which they sit.

### Thursday 20th October

There will be no seminar this week.

### Thursday 27th October

Marc Lackenby (Oxford)

Counting covering spaces and subgroups in dimension three

### Thursday 3rd November, 13:30

Thierry Monteil (IML, Marseille)

Finite blocking property on polygonal billiards and translation surfaces

### Thursday 3rd November, 15:00

Hideki Miyachi (Tokio Denki)

On the image of the asymptotic map

### Thursday 10th November

Samuel Lelièvre (Warwick)

Square-tiled surfaces: telling orbits apart and counting

Abstract: There is a natural action of the group SL(2,R) on strata of moduli spaces of abelian differentials, whose orbits are Teichmüller discs. We will discuss how to enumerate the Teichmüller discs of square-tiled surfaces and how to count their integer points in the case of abelian differentials with a double zero on Riemann surfaces of genus two.

### Thursday 17th November

Jens Marklof (Bristol)

Geometric aspects of the distribution of n^2 alpha mod 1

Abstract: It is well known that for every irrational alpha, the sequence n^2 alpha is equidistributed modulo one: the number of n=1,...,N for which the fractional part of n^2 alpha is contained in a subinterval [a,b] of the unit interval is approximately N(b-a) for N large. I will explain, how more subtle properties of the sequence that characterize its randomness can be described in terms of geodesics on the modular surface. One particular application is a limit law for skew translations.

### Thursday 24th November

Bertrand Deroin (IHES)

Random 1-dimensional dynamical systems

Abstract: This is a joint work with Victor Kleptsyn. We study the Brownian motion along the leaves of a foliation. This theory were originated by L. Garnett. Here we restrict our attention to the foliations that are transversely conformal, for instance real codimension $1$ foliations or complex codimension $1$ holomorphic foliations. We prove that either there is a transversely invariant measure, or there is a transverse contraction phenomenom and a unique invariant measure by the leafwise Brownian motion. The proof deals with the Lyapunov exponents of such invariant measures.

### Thursday 19th January, Room B3.01

Cormac Long (Southampton)

Some dodecahedral tessellations associated with the Coxeter group $[5,3,5]$

Abstract: We describe some dodecahedral tessellations of hyperbolic $3$-manifolds arising from subgroups of the Coxeter group $[5,3,5]$. These include a chiral pair of copies of the Weber-Seifert space, their homology covers and their smallest common cover. We also discuss some other covering manifolds that arose during this study.

### Thursday 26th January

Bas Lemmens (Warwick)

How many points at the same distance?

Abstract: A set in a metric space is called equilateral (or equidistant) if any two distinct elements are at the same distance from each other. It is well known that an equilateral set in the n-dimensional Euclidean space has at most n+1 elements. But how large can such a set be if the norm is changed to another norm say an $\ell_p$ norm? In this talk I will discuss several results and conjectures concerning this problem.

### Thursday 2nd February

Moon Duchin (UC Davis)

Curvature and dynamics in Teichmuller space

Abstract: Standard notions of nonpositive curvature do not hold for the Teichmuller metric, so we introduce a new notion which does capture some of the hyperbolic behavior in that metric. Using this condition, we show that the random action of the mapping class group is well approximated by the geodesic flow.

### Thursday 9th February

Mikhail Belolipetsky (Durham)

Finite groups and hyperbolic manifolds

Abstract: The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg and Kojima, respectively. Our proof is non constructive: it uses counting results from subgroup growth theory and the strong approximation theorem to show that such manifolds exist. This is joint work with Alex Lubotzky.

### Thursday 16th February - POSTPONED

Caroline Series (Warwick)

Lines of minima and Teichmüller geodesics I

### Thursday 23rd February - POSTPONED

Caroline Series (Warwick)

Lines of minima and Teichmüller geodesics II

### Thursday 2nd March

James Giblin (Warwick)

Classification of continuously transitive Circle Groups

A group of homeomorphisms G acting on a topological space X is continously transitive if given a pair of continuous paths f, g:[0,1]\to X, there exists a continuous path of homeomorphisms F:[0,1]\to G satisfying F(t)(f(t))=g(t). In recent work with Vladimir Markovic, we have classified all such groups when the space X is S^{1}. This work also settles two conjectures of Ghys and de la Harpe. In this talk I will explain this classification in more detail and give an overview of the proof.

### Monday 6th March, 11am, Room B3.02

Brian Bowditch (Southampton)

The Ending Lamination Conjecture I

### Wednesday 8th March, 4pm, Room B3.01

Brian Bowditch (Southampton)

The Ending Lamination Conjecture II

### Thursday 9th March, 1:30pm, Room B3.02

Brian Bowditch (Southampton)

The Ending Lamination Conjecture III

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

Pascal Hubert

Veech groups and the dynamics of flat surfaces

To study linear flows on translation surfaces, it is very useful to consider the SL(2,R) action on the parameter space (a stratum of the moduli space of abelian differentials). In 1989, Veech proved a celebrated result: if the stabilizer of the SL(2,R) orbit of a translation surface is a lattice (a Fuchsian group of finite covolume) then the dynamics on the flat surface is optimal, i.e., linear flows are periodic or uniquely ergodic. These stabilizers are now called Veech groups. In this talk, we will give several constructions of Veech groups: infinitely generated Veech groups, Veech groups without parabolic elements, etc.

### Thursday 30th March, 15:00, Room B3.02

Emmanuel Royer

Quasimodular forms and applications

The aim of this talk is to introduce the notion of quasimodular form and to explain its importance. These functions show up in number theory as well as in geometry and in physics. Their introduction, due to Rankin and to Kaneko & Zagier, answers a natural question in number theory, that of taking into account derivatives of modular forms. Their importance in geometry, notably in counting torus coverings, has been emphasised by Dijkgraaf, then by Eskin, Masur & Schmoll and by Bloch & Okounkov.

### Thursday 20th April

Jack Button (Cambridge)

3-manifold groups, the BNS invariant and the Alexander polynomial

### Thursday 27th April, 11:00, Room B3.02 ***CANCELLED***

Anthony O'Farrell (NUI, Maynooth)

Reversibility and its Connections

A reversible map is one that is conjugate to its inverse. Such maps appear naturally in classical dynamics, for instance in the pendulum, the n-body problem or billiards. They also arise in less obvious ways in connection with problems in geometry, complex analysis, approximation, and functional equations. When a problem has a connection to a reversible map, this opens it to attack using dynamical ideas, such as ergodic theory and the theory of flows. We'll discuss an example or two. It would be interesting to understand reversibility better. We'll discuss this problem, and a little progress. The subject relates to involutions, and to conjugacy problems, in classical groups, groups of power series, and larger groups. Some results are joint work with Maria Roginskaya, Ian Short, and Roman Lavicka.

### Thursday 27th April, 13:30, Room B3.02

Etienne Ghys (ENS Lyon)

Rotation numbers for surface automorphisms

Given an area preserving diffeomorphism of a compact surface, one would like to introduce dynamical invariants which measure the "amount of rotation of the differential". Some of these invariants have been introduced long time ago by Calabi and Ruelle. Others have been introduced much more recently in particular by Entov-Poletrovich, Gambaudo-Ghys and Py. I would like to survey these constructions and try to compare them.

### Thursday 4th May

Quasiconformally homogenous surfaces

### Thursday 11th May

Greg McShane (Toulouse)

The Hitchin component, higher Teichmueller space and shear coordinates

### Thursday 18th May - Joint Algebra/Geometry Seminar

Susan Hermiller (University of Nebraska - Lincoln)

Almost convexities and tame combings for groups

In this talk I will discuss the geometry of the Baumslag-Solitar groups. This will be used to illustrate the relationships between various almost convexity properties and tame combings. This is joint work with Sean Cleary.

### Thursday 25th May

Martin Bridson (Imperial College)

Snowflake groups, Perron-Frobenius exponents and isoperimetric spectra

### Thursday 1st June

Hugo Parlier (University of Geneva)

Gaps left by simple closed geodesics on surfaces

This talk concerns joint work with Peter Buser. For a Riemann surface endowed with a hyperbolic metric, J. Birman and C. Series have shown that the set of all points lying on any simple closed geodesic is nowhere dense on the surface. (This set is sometimes referred to as the Birman-Series set). The talk will discuss the existence of positive constants $C_g$, such that for any surface of genus $g$, the complementary region to the Birman-Series set allows an isometrically embedded disk with radius $C_g$. The behavior of $C_g$ in function of $g$, as well as some bounds will also be discussed.

### Thursday 8th June

Robert Silhol (Montpellier)

Hyperbolic Lego, translation surfaces and uniformization

### Thursday 15th June

Frank Herrlich (Karlsruhe)

Equations for Teichmüller curves: some examples

If a Teichmüller geodesic in Teichmüller space descends to an algebraic curve in moduli space, this image is called a Teichmüller curve. In the talk I shall present a few examples (all coming from origamis) where the corresponding 1-parameter family of Riemann surfaces can be described explicitly by an algebraic equation.

### Thursday 22nd June, 13:00, Room B1.01

Anthony O'Farrell (Maynooth)

Reversibility and its Connections

A reversible map is one that is conjugate to its inverse. Such maps appear naturally in classical dynamics, for instance in the pendulum, the n-body problem or billiards. They also arise in less obvious ways in connection with problems in geometry, complex analysis, approximation, and functional equations. When a problem has a connection to a reversible map, this opens it to attack using dynamical ideas, such as ergodic theory and the theory of flows. We'll discuss an example or two. It would be interesting to understand reversibility better. We'll discuss this problem, and a little progress. The subject relates to involutions, and to conjugacy problems, in classical groups, groups of power series, and larger groups. Some results are joint work with Maria Roginskaya, Ian Short, and Roman Lavicka.

### Thursday 29th June ***CANCELLED***

Akshay Venkatesh (Currently visiting the Institute for Advanced Study, Princeton)

TBA

Current year of the seminar.