# Geometry and Topology Seminar

## Warwick Mathematics Institute, Term III, 2014-2015

 Thursday April 9, 15:00, room MS.04 Martin Kassabov (Cornell) Hopf algebras and invariants of the Johnson cokernel Abstract: We show that if H is a cocommutative Hopf algebra, then there is a natural action of $\Aut(F_n)$ on $H^n$ which induces an $\Out(F_n)$ action on a quotient. In the case when $H = T(V_{2g})$ is the tensor algebra, we show that there is a surjection from the cokernel of the Johnson homomorphism for the mapping class group of a surface of genus $g$ to the coholomogy groups $H^{\vcd}(\Out(F_n))$. We calculate the $n = 2$ case, getting large families of previously unknown obstructions to the image of the Johnson homomorphism.

 Thursday May 7, 15:00, room MS.04 John Parker (Durham) Non-arithmetic lattices Abstract: A lattice in a semi-simple Lie group $G$ is a discrete subgroup whose quotient has finite Haar measure. It acts on the associated symmetric space as a discrete group of isometries with finite covolume. An arithmetic subgroup of a linear algebraic group is a subgroup that is discrete because the integers are discrete in the real numbers. All lattices are arithmetic except when $G$ is $\SO(n,1)$ or $\SU(n,1)$. The cases of $\SO(n,1)$ and $\SU(1,1)$ are quite well understood. There are nine examples (up to commensurability) of non-arithmetic lattices in $\SU(2,1)$ and there is a single example in $\SU(3,1)$. These are all due to Deligne and Mostow in 1986. For $n$ at least four the problem is open. In joint work with Deraux and Paupert we have found ten more examples of non-arithmetic lattices in $\SU(2,1)$, the first to be found since 1986. I will give a gentle introduction to the problem and outline how we found our new examples.

 Thursday May 14, 15:00, room MS.04 Richard Webb (UCL) Describing loops in surfaces and 3--manifolds, and applications Abstract: The curve graph of a surface has played a vital role in our understanding of infinite volume hyperbolic 3--manifolds and mapping class groups. In this talk we shall explain how Masur and Minsky's hierarchy machinery has been important, and discuss how effective hierarchies are for explicitly describing the mapping class group and hyperbolic 3--manifolds in terms of the underlying surface. (This is joint work with Tarik Aougab and Samuel Taylor.)

 Thursday June 4, 15:00, room MS.04 Jenya Sapir (UIUC) Counting non-simple closed curves on surfaces Abstract: We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result.

 Thursday June 11, 15:00, room MS.04 Catherine Pfaff (Bielefeld) When outer space behaves like Teichmüller space (or hyperbolic spaces) and how we can use this to understand $\Out(F_r)$ Abstract: $\Out(F_r)$ is one of the most intriguing groups to study because of its natural action on a space, Culler-Vogtmann outer space, which both strongly resembles and intricately differs from some of the most well-known and studied spaces, such as Teichmüller space and hyperbolic spaces. In this talk I will present several dynamical results about when outer space behaves like these other spaces and explain how we have used them to help understand $\Out(F_r)$. This is joint work with Yael Algom-Kfir, Ilya Kapovich, and Lee Mosher.

 Thursday June 18, 15:00, room MS.04 Beatrice Pozzetti (Warwick) Bounded cohomology and applications Abstract: Bounded cohomology was introduced by Gromov. Despite its definition being rather similar to that of singular cohomology, it is very difficult to compute and is still poorly understood. After recalling the definition of bounded cohomology of spaces and of groups I will mention some new results for graphs of groups (for example free or amalgamated products) and acylindrically hyperbolic groups (for example the mapping class group). In the second part of the talk I will discuss my favourite application of bounded cohomology: maximal representations. Burger, Iozzi, and Wienhard used bounded cohomology to select some components of the space of homomomorphisms from the fundamental group of a surface to a semisimple Lie group of Hermitian type (for example $\Sp(2n,\RR)$) which should be thought of as higher rank analogues of the Teichmüller space. I will motivate why they are interesting objects to study, describe some geometric features of maximal representations, and maybe present a rigidity result. The results referred to in the first part are joint work with Bucher-Burger-Iozzi-Frigerio-Pagliantini and Frigerio-Sisto respectively.

 Thursday June 25, 16:30, room MS.05 Mark Sapir (Vanderbilt) Aspherical Higman embeddings Abstract: I will show how to embed every finitely generated group with two-dimensional recursive $K(\cdot,1)$ into a group with finite 2-dimensional $K(\cdot,1)$. This implies (via Davis' trick and Gromov's construction of random groups) that there exists a four-dimensional compact Riemannian manifold whose fundamental group contains an expander, hence does not coarsely embed into a Hilbert space, has infinite asymptotic dimension, does not satisfy the Baum-Connes conjecture with coefficients, etc.

Information on past talks. This line was last edited Tuesday, 7 April 2015 15:20:10 BST.