A classical area of study in geometric group theory is subgroup growth, which counts the number of subgroups of a given group Gamma as a function their index. We will study a richer function, the commensurability growth, which is a function associated to a subgroup Gamma in an ambient group G. This talk covers the case that Gamma is an arithmetic subgroup of a unipotent group G, starting with the simplest example of the integers in the real line. This is joint work with Khalid Bou-Rabee.

A classical theorem of Powell (with roots in the work of Mumford and Birman) states that the pure mapping class group of a connected, orientable, finite-type surface of genus at least 3 is perfect, that is, it has trivial abelianization. We will discuss how this fails for infinite-genus surfaces and give a complete characterization of all homomorphisms from pure mapping class groups of infinite-genus surfaces to the integers. This is joint work with Javier Aramayona and Nicholas Vlamis.

I will discuss some joint work with Aramayona, Ghaswala, Kent, McLeay, and Winarski on the Torelli subgroup of big mapping class groups.

Bounded cohomology admits an almost entirely algebraic definition (which we give in the talk), but many examples come from geometric constructions. Any isometric action of a discrete group G on hyperbolic n-space defines a class in bounded cohomology which measures the hyperbolic volume of geodesic simplices whose vertices are contained in an orbit G.x. We explain a sufficient condition, phrased in terms of the discrete subgroups of Isom(H^n) for the volume class of a dense action to yield a non-zero class in bounded cohomology.

In this talk we will focus on our joint work in progress with Daniele Alessandrini and Anna Wienhard about quasi-Hitchin representations in Sp(4,C), which are deformation of Fuchsian representations. We will show that the quotient of the domain of discontinuity for the action of these representations on the space of complex lagrangians Lag(C^4) is a fiber bundle over the surface and we describe the fiber. In particular, we will describe how the projection comes from an interesting parametrization of the space Lag(C^4) of complex lagrangians as the space of regular ideal hyperbolic tetrahedra and their degenerations.

In the mapping class group of a surface, stabilisers of curves are quasi-isometrically embedded for all curves. In this talk, we study a similar question for mapping class groups of 3-dimensional handlebodies and find a different picture (analogous to the situation in Out(F_n)): stabilisers of meridians are quasi-isometrically embedded, while stabilisers of other curves may be exponentially distorted.

The theory of Diophantine approximation is underpinned by Dirichlet’s fundamental theorem. Broadly speaking, the main questions in the theory concern quantifying the prevalence of points with exceptional behavior with respect to Dirichlet’s result. Badly approximable, very well approximable and Dirichlet-improvable points are among the most well-studied such exceptional sets. The work of Dani and Kleinbock-Margulis connects these questions to the recurrence behavior of certain flows on homogeneous spaces. After a brief overview of the subject and the motivating questions, I will discuss new results giving a sharp upper bound on the Hausdorff dimension of divergent orbits of diagonal flows emanating from fractals on the space of lattices. Applications towards Diophantine approximation will be presented.

The first occurrence of topology in the study of fibred surfaces is the Zeuthen-Segre formula, which I shall recall; the second one is a consequence of the Castelnuovo De Franchis theorem, which shows the topological nature of maps to curves of genus at least two. Kodaira fibrations illustrate the principle that the index is not multiplicative for differentiable fibre bundles, and the theorems of Fujita and Arakelov relate this property to Variation of Hodge structure. There are important open questions on fibred surfaces, one of them being the Shafarevich conjecture. I might mention, starting from the BCD surfaces which provide examples where the monodromy on the flat unitary bundle in the Fujita decomposition is of infinte order, some criterion which ensures that the universal covering of a fibred surface S is contractible. I shall then discuss joint work with Corvaja and Zannier, and partial results concerning commutators in the symplectic group and in the mapping class group: these results are relatively exhaustive concerning the symplectic group. These questions arose from the question of understanding the singular fibres in the somewhat mysterious Cartwright-Steger. I shall then produce a simple example, of a fibration of the product of two curves of genus 2 (or more) onto an elliptic curve, with only one irreducible singular fibre. This shows that the product of 4 homologically independent Dehn twists is a commutator in Map_g for g at least 9.

Poincaré's recurrence theorem says that if (X,mu) is a probability measure space and T: X -> X is a transformation preserving mu, then for every subset A of X having positive measure, there is a natural number n such that the sets A and T^-n(A) have nonempty intersection. For a fixed small number c, we consider the set R of times n when this intersection has measure at least c. Such sets of "large intersection times" have rich combinatorial structure. In particular they contain perfect squares, a striking fact proved independently by Furstenberg and A. Sárközy. We will construct sets of large intersection times having (or lacking) certain interesting features. The corresponding transformations have will large rigidity sequences: sequences n_k along which the powers T^n_k approach the identity transformation in a natural sense. Our construction relies on Kronecker sets, a class of thin subsets of the unit interval first studied in classical harmonic analysis.

Flat cone metrics on surfaces (often in the guise of translation surfaces or holomorphic 1-forms) are a fundamental object of study in Teichmüller theory, billiard dynamics, and complex geometry. Fixing the number and angle of the cone points defines a natural subvariety of the moduli space of flat surfaces called a stratum, the global topology of which is quite enigmatic. In this talk, I will explain which mapping classes are realized by deformations contained in these strata, and how this result can be applied to classify the connected components of Teichmüller spaces of flat cone metrics. This talk represents joint work with Nick Salter.

Various questions concerning translation surfaces depend on counting saddle connections. For a certain class of translation surfaces, this reduces to the more general, yet more tractable problem of counting points in discrete orbits for the linear action of a lattice of SL(2,R) on the Euclidean plane. This can be done effectively, using either methods from ergodic theory or from number theory. We will illustrate both methods and discuss certain of their aspects. This is joint work with Amos Nevo, René Rühr, and Barak Weiss.