# Geometry and Topology

Please contact Beatrice Pozzetti or Saul Schleimer if you would like to speak or suggest a speaker.

## Term 2

 Thursday January 19, 16:00, room MS.03 Dusa McDuff (Columbia) Symplectic topology today Colloquium: This talk will explain the basics of symplectic topology for a nonspecialist audience, outlining some of the classical results as well as some problems that are currently open.

 Thursday January 26, 15:00, room MS.03 Marc Burger (ETH Zurich) On finiteness properties of universal groups Abstract: Given a regular tree T of valency n and a permutation group F on n elements, one can associate the closed subgroup U(F) of those automorphisms of T which act locally like elements of F. These groups have been introduced in joint work with S.Mozes in the context of the study of lattices in the product of two trees. A major open problem is the characterization of F such that U(F) is the closure of the projection of a cocompact lattice in the product of two regular trees TxT'. A necessary condition is that open compact subgroups of U(F) be topologically finitely generated; in this talk we'll report on joint work with S. Mozes characterizing the F's for which U(F) has his property.

 Thursday February 2, 15:00, room MS.03 Nicolas Tholozan (ENS Paris) $\mathrm{PSL}(2,\mathbb{R})$-Representations of the fundamental group of the punctured sphere Abstract: A famous question of Bowditch asks whether a (type-preserving) representation of the fundamental group of a (punctured) surface into $\mathrm{PSL}(2,\mathbb{R})$ which is not Fuchsian must send a simple closed curve to an elliptic or parabolic element. In this talk, I will show that some interesting representations of the fundamental group of the $n$-punctured sphere have an even stronger property : they map \emph{every} simple closed curve to an elliptic or parabolic element. We will show that these representations form compact components of relative character varieties which are symplectomorphic to $\mathbb{C} \mathbf{P}^{n-3}$ with a multiple of the Fubini--Study symplectic form. This is a joint work with Bertrand Deroin.

 Thursday February 9, 15:00, room MS.03 Ingo Blechschmidt (Augsburg) A leisurely introduction to synthetic differential geometry Abstract: A tangent vector is an infinitesimal piece of a curve." Pictures likes this are routinely used when thinking informally about differential geometry, but they are not literally true in the usual setup. Synthetic differential geometry, iniated by Anders Kock, provides a different approach to differential geometry in which such statements are literally true. This is accomplished by switching to an alternate topos, a mathematical universe, in which the real numbers contain infinitesimal elements -- numbers $\varepsilon$ such that $\varepsilon \neq 0$ but $\varepsilon^2 = 0$. In this way it's possible to interpret, for instance, some of Sophus Lie's writings literally. Additionally synthetic differential geometry enables some new modes of thought. Differential forms, for example, are classically functionals on tangent vectors. In synthetic differential geometry, it's also possible to think about differential forms as quantities, that is functions defined on the manifold. Ideas related to synthetic differential geometry recently allowed Oliver Fabert to prove a version of the Arnold conjecture in infinite dimensions. The talk gives a leisurely introduction to synthetic differential geometry, presenting the synthetic definitions and establishing the link to the usual setup of differential geometry. No prior knowledge about toposes or formal logic is supposed.

 Thursday February 23, 15:00, room MS.03 Ashot Minasyan (Southampton) On conjugacy separability of subdirect products Abstract: Let $\mathcal{C}$ be a class of groups (e.g., all finite groups, all $p$-groups, etc.). A group $G$ is said to be {\it residually-$\mathcal C$} if for any two distinct elements $x,y \in G$ there is a group $M \in \mathcal{C}$ and a homomorphism $\varphi: G \to M$ such that $\varphi(x) \neq \varphi(y)$ in $M$. Similarly, $G$ is $\mathcal C$-conjugacy separable if for any two non-conjugate elements $x,y \in G$ there is $M \in \mathcal{C}$ and a homomorphism $\varphi:G \to M$ such that $\varphi(x)$ is not conjugate to $\varphi(y)$ in $M$. When $\mathcal{C}$ is the class of all finite groups, the above \emph{residual properties} are closely related to the two main decision problems in groups: the word problem and the conjugacy problem. In the talk I will discuss $\mathcal C$-conjugacy separability of subdirect products $G \leq F_1 \times F_2$, where $F_i$, $i=1,2$, are either free or hyperbolic (in the sense of Gromov). Recall that a subgroup $G$ of a direct product of two groups $F_1\times F_2$ is said to be \emph{subdirect} if $G$ projects onto each of the coordinate groups $F_1,F_2$. In this case, it is easy to see that $N_1:=F_1 \cap G$ is a normal subgroup of $F_1$. Classically, in Combinatorial Group Theory, sudirect (fibre) products have been used to produce examples of groups with exotic properties. The standard underlying idea, originating from Mihajlova's trick, is that ''bad'' properties of the quotient $F_1/N_1$ transfer to ''less bad'' properties of the subdirect product $G$, and ''good'' properties of $F_1/N_1$ give rise to ''even better'' properties of $G$. Following this philosophy, we will prove that if $F_1/N_1$ is not residually-$\mathcal C$ then $G$ is not $\mathcal{C}$-conjugacy separable; on the other hand, if $F_i$, $i=1,2$, are free and all cyclic subgroups of $F_1/N_1$ are closed in the pro-$\mathcal{C}$-topology, then $G$ is $\mathcal{C}$-conjugacy separable. These criteria can be used to produce examples of subdirect products of free/hyperbolic groups which are conjugacy separable, but have non-conjugacy separable subgroups of finite index and vice-versa. Other applications will also be discussed.

 Thursday March 2, 15:00, room MS.03 Henry Wilton (Cambridge) Surface subgroups of graphs of free groups Abstract: A well known question, usually attributed to Gromov, asks whether every hyperbolic group is either virtually free or contains a surface subgroup. I’ll discuss some recent progress on this problem for a the class of groups in the title.

 Monday, March 9, 15:00, room MS.03 Saul Schleimer (Warwick) Circular orderings from veering triangulations Abstract: This is joint work with Henry Segerman. Suppose that (M,T) is a cusped hyperbolic three-manifold equipped with a veering triangulation. We show that there is a unique circular order on the cusps of the universal cover of M, which is compatible with T. After giving the necessary background and sketching the proof, I will speculate wildly about possible applications.

 Thursday March 16, 15:00, room MS.03 Marco Golla (Uppsala) Dehn surgery and rational balls Abstract: Dehn surgery is a very natural cut-and-paste operation on 3-manifolds. In this talk I will give restrictions on when this operation yields 3-manifolds bounding the 'smallest' possible 4-manifolds, namely rational balls. Depending on time, I will discuss relations to singular complex curves or the problem of embedding 3-manifolds in 'small' 4-manifolds. This is mostly joint work with Paolo Aceto.

## Term 1

 Thursday October 6, 15:00, room MS.03 Antonio De Capua (Oxford) Train tracks, pants graphs, and hyperbolic volume Abstract: Given a topological surface $S$, a number of graphs representing patterns of curves on $S$ are naturally constructed. The marking graph is a graph whose vertices, called markings, are families of curves on $S$ which cut it into topological discs; and there is an edge between two markings every time they are related to each other via a pre-defined elementary move. The pants graph of $S$ is built similarly, but its vertices are pants decompositions on $S$. The study of these graphs is far from simple, but the so-called train tracks i.e. pictures on $S$ resembling railway networks, have proven to be a useful way of modelling combinatorially families of curves, thus helping in this task. A reiteration of certain elementary moves on a train track produces what is called a splitting sequence: Masur, Mosher and Schleimer have shown how splitting sequences may be used to get distance estimates in the marking graph. I will describe how to adapt their approach to the pants graph. This requires a tailored version of this latter graph and a manipulation, described algorithmically, of the given splitting sequence. The presented result has an interesting application in estimating the volume of a class of hyperbolic 3-manifolds: namely pseudo-Anosov mapping tori, via a theorem of Brock and the maximal splitting sequence introduced by Agol.

 Thursday October 13, 15:00, room MS.03 Mark Bell (UIUC) Automorphisms of the polygonalisation complex Abstract: We will discuss a new complex associated to a surface: the polygonalisation complex. This records the different ways to decompose the surface into polygons and sits somewhere between the arc complex and the flip graph. We will show that this is a "rich" complex in that its automorphism group is the mapping class group of the surface. Thus this complex joins a long list of complexes, including the arc complex, pants graph, curve complex and flip graph, that satisfy Ivanov's meta-conjecture. This is joint work with Valentina Disarlo and Robert Tang.

 Thursday October 20, 15:00, room MS.03 Brian Bowditch (Warwick) Rigidity of complexes associated with a surface Abstract: There are lots of different complexes one can associate with a compact surface, and these have found many applications. In particular, many of these complexes are rigid, in the sense that any automorphism is induced by an element of the mapping class group. We will give a brief exposition of some of these results, and how they can be applied.

 Thursday October 27, 15:00, room MS.03 David Martí Pete (The Open University) The escaping set of transcendental self-maps of the punctured plane Abstract: We study the iteration of holomorphic self-maps of C*, the complex plane with the origin removed, for which both zero and infinity are essential singularities. The escaping set of such maps consists of the points whose orbit accumulates to zero and/or infinity following what we call essential itineraries. We show that the Julia set always contains escaping points with every essential itinerary. The concept of essential itinerary leads to a partition of the escaping set into uncountably many disjoint sets, the boundary of each of which is the Julia set. Under certain hypotheses, each of these sets contains uncountably many curves to zero and infinity. We also use approximation theory to provide examples of functions with escaping Fatou components.

 Thursday November 3, 15:00, room MS.03 Luca F. Di Cerbo (ICTP) On the volume spectrum of ball quotient surfaces Abstract: I will construct two infinite families of noncompact complex hyperbolic 2-manifolds whose volume spectrum is the set of all integral multiples of 8/3\pi^{2}, i.e., they both saturate the volume spectrum of ball quotient surfaces. The surfaces in one of the two families have all 2-cusps, so that one can saturate the entire volume spectrum with 2-cusped manifolds. Finally, I show that the associated neat lattices have infinite abelianization and finitely generated commutator subgroup. These appear to be the first known nonuniform lattices in PU(2, 1), and the first infinite tower, with this property. Joint work with M. Stover.

 Thursday November 10, 15:00, room MS.03 Andre de Carvalho (Stony Brook) Low-dimensional dynamics and hyperbolic 3-manifolds Abstract: Thurston’s hyperbolization of fibered 3-manifolds is based on his classification theorem for isotopy classes of surface homeomorphisms. This classification has also been extremely important to the study of dynamical systems on surfaces. The classification theorem holds for surfaces of finite topological type, but in dynamics one is always interested in infinities: infinite time, infinite orbits, etc. In this talk we discuss some specific instances of this contrast. We introduce generalized pseudo-Anosov maps, which are pA maps which are allowed to have infinitely many singularities. We then discuss the possibility of hyperbolizing the associated mapping tori and some similarities with the Otal proof of hyperbolization. We also discuss families of mapping tori associated to families of unimodal maps, limits of certain sequences of hyperbolic 3-manifolds within these families and relate this to the previous hyperbolization discussion.

 Thursday November 17, 15:00, room MS.03 Hugo Parlier (Fribourg) Interrogating length spectra of surfaces Abstract: Associated to a closed hyperbolic surface is its length spectrum, the set of the lengths of all of its closed geodesics. Two surfaces are said to be isospectral if they share the same length spectrum. There are different methods to produce surfaces that are isospectral but not isometric, the most successful one based on a technique introduced by Sunada. This talk will be about giving quantitative answers to the following questions: - How many “questions" do you need to ask to determine a length spectrum? - In a given genus how many different surfaces can be isospectral but not isometric? The approach will include finding adapted coordinate sets for moduli spaces and exploring McShane type identities.

 Thursday November 24, 15:00, room MS.03 Stefano Riolo (Pisa) Hyperbolic Dehn filling in dimension four Abstract: By gluing copies of a deforming polytope, we describe some deformations of complete, finite-volume hyperbolic cone four-manifolds. Despite the fact that hyperbolic lattices are locally rigid in dimension greater than three (Garland-Raghunathan), we see a four-dimensional analogue of Thurston's hyperbolic Dehn filling: a path of cone-manifolds M_t interpolating between two cusped hyperbolic four-manifolds M_0 and M_1. Joint work with Bruno Martelli.

 Monday, December 12, 14:00, room MS.04 Benjamin Miesch (ETH Zurich) The Cartan-Hadamard Theorem for Metric Spaces with Local Geodesic Bicombings Abstract: Local-to-global principles are spread all-around in mathematics. The classical Cartan-Hadamard Theorem from Riemannian geometry was generalized by W. Ballmann for metric spaces with non-positive curvature, and by S. Alexander and R. Bishop for locally convex metric spaces. We prove the Cartan-Hadamard Theorem in a more general setting, namely for spaces which are not uniquely geodesic but locally possess a suitable selection of geodesics, a so-called convex geodesic bicombing. Furthermore, we deduce a local-to-global theorem for injective (or hyperconvex) metric spaces, saying that under certain conditions a complete, simply-connected, locally injective metric space is injective.

Information on past talks.

Internet Access at Warwick:
Where possible, visitors should obtain an EDUROAM account from their own university to enable internet access whilst at Warwick.
If you need WiFi whilst at Warwick, click here for instructions (upon arrival at Warwick)
Registration:
You can register for any of the symposia or workshops online. To see which registrations are currently open and to submit a registration, please click here.
Contact:
Mathematics Research Centre
Zeeman Building
University of Warwick
Coventry CV4 7AL - UK
E-mail:
mrc@maths.warwick.ac.uk