
# Geometry and Topology Seminar

## Warwick Mathematics Institute, Term II, 2016-2017

 Thursday January 19, 16:00, room B3.02 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 Zuerich) 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) 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.

 Thursday 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) Abstract:

Information on past talks. This line was last edited Tuesday, January 19, 2016 08:35:36 PM GMT