The curve graph is a locally infinite graph. In this talk, we will show that the curve graph could be understood as a locally finite graph via subsurface projections. As an application, we obtain an uniform bound of Bowditch's slices of the collection of all tight geodesics between any given pair of curves.

The graph braid group over a given graph is the fundamental group of the configuration space over the graph. We will discuss presentations and the first homologies of graph braid groups with historical facts.

In 1992, Alan Reid proved that if two arithmetic hyperbolic 2-manifolds have the same geodesic length spectrum, the two manifolds must be commensurable. In 2008, Chinburg-Hamilton-Long-Reid extended Reid's result to arithmetic hyperbolic 3-manifolds. In this talk, I will discuss effective versions of these results. Specifically, given two arithmetic hyperbolic 2- or 3-manifolds of some bounded volume V that are not commensurable, we ensure that a length L occurs in one but not both. More important, the length L can be bounded above as a function of the volume V and is explicitly given. These results rely on effective rigidity results for quaternion algebras. The main tools used are algebraic and geometric counting results of independent interest. Time permitting, I will discuss some of these counting results. This work is joint with Benjamin Linowitz, Paul Pollack, and Lola Thompson.

An important class of dynamical systems are interval exchange transformations. One cuts up an interval and rearranges the pieces by translations. In the case of 2 intervals this is equivalent to a rotation of a circle. If there are 4 or more intervals an interesting phenonemom can happen that the the transformation is minimal but not uniquely ergodic. This means that there is more than one invariant measure. The set of invariant probability measures is a convex set and the ergodic measures are the extreme points. The Birkhoff theorem says that for an ergodic measure almost every point is generic whch means that its orbit is uniformly distributed. A question is whether a nonergodic measure can have such generic points. In this talk I answer this question affirmatively with an example. This is joint work with Jon Chaika.

We provide a bi-combing of Teichmüller space equipped with the Teichmüller metric by modifying the Teichmüller geodesic paths. As corollary, we conclude that the Dehn functions of the Teichmuller space are Euclidean.

In this talk we will show that a planar graph is the1-skeleton of a Euclidean polyhedron inscribed in a hyperboloid if and only if it is the 1-skeleton of a Euclidean polyhedron inscribed in a cylinder if and only if it is the 1-skeleton of a Euclidean polyhedron inscribed in a sphere and has a Hamiltonian cycle. This result follows from the characterisation of ideal polyhedra in anti-de Sitter and half-pipe space in terms of their dihedral angles and induced metric on its boundary. (This is joint work with J Danciger and J-M Schlenker.)

The curve and pants graphs of a surface S encode information about hyperbolic structures on 3-manifolds fibering over the circle with fiber S. However, the way in which the geometry of these graphs explicitly depend on S is not well understood, and this limits our ability to use these graphs to make concrete statements about 3-manifolds. In this talk we will discuss a collection of results which shed light on this dependence. As an application, we give an effective version of Brock's coarse volume formula, which relates the volume of a hyperbolic 3-manifold fibering over the circle with fiber S, to the translation length of the monodromy acting on the pants graph. This is joint work with Samuel Taylor and Richard Webb.

The palindromic automorphism group of a free group is the group of automorphisms that take each member of some fixed free basis to a word that reads the same backwards as forwards. This group is an obvious free group analogue of the hyperelliptic mapping class group of an oriented surface. I will discuss some elementary properties of palindromes and palindromic automorphisms, and introduce a new complex on which the palindromic automorphism group acts. In particular, we will discuss how the action on this complex can be used to find a generating set for the so-called palindromic Torelli group. I will also discuss recent joint work with Anne Thomas on generalisations of these results to the right-angled Artin group setting.

I will survey some recent developments in the theory of flat surfaces of finite area and translation flows, including both compact and (infinite genus) non-compact surfaces. In particular, I will concentrate on a new point of view based on a joint paper with K. Lindsey, where we develop a close connection of Bratteli diagrams and flat surfaces. I will also state a criterion for unique ergodicity in the spirit of Masur's criterion which holds in this very general setting and which implies Masur's criterion in moduli spaces of (compact) flat surfaces. No knowledge of anything will be assumed, and the talk will not be technical and full of examples

When does Borel's theorem on free subgroups of semisimple groups generalize to other groups? In this talk, we present a systematic study of this question and arrive at positive and negative answers for it. In particular, we find a full classification of fundamental groups of surfaces and von Dyck groups that satisfy Borel's theorem. We also discuss connections with a question of Breuillard, Green, Guralnick, and Tao concerning double word maps.

Let G be a group equipped with a finite generating set S. G is called growth tight if its exponential growth rate relative to S is strictly greater than that of every quotient G/N with N infinite. This notion was first introduced by Grigorchuk and de la Harpe. Examples of groups that are growth tight include free groups relative to bases and, more generally, hyperbolic groups relative to any generating set. In this talk, I will provide some sufficient conditions for growth tightness which encompass all previous known examples.

The full residual finiteness growth (FRF growth) of a group G measures the difficulty of detecting word metric balls in G using finite quotients of G. This is one kind of quantification of the residual finiteness of G. We will discuss FRF growth of nilpotent groups, with the goal of proving that FRF growth is precisely polynomial in the class of nilpotent groups. This talk covers work joint with Khalid Bou-Rabee.

We'll talk about a nonnegative curvature result for higher rank arithmetic groups.

We will discuss a recent result of Kojima-McShane relating the hyperbolic volume of a 3-manifold that fibers over the circle to the translation distance of the corresponding pseudo-Anosov diffeomorphism.

We will describe ongoing work with J. Athreya on the distribution of lattice points of a random lattice chosen uniformly with respect to Haar measure on the space of Heisenberg lattices. The main question we will discuss is: 'Given a set of positive measure in three dimensional Euclidean space, what is the probability that a random Heisenberg lattice will not hit the set?' This work extends results of Athreya and Margulis who settled the Euclidean case and is part of a broader program to understand statistics of lattices in a wide variety of algebraic and Lie groups. Our methods combine the orbit structure of certain actions of the special linear group and the continuous part of the spectral decomposition on the space of Euclidean lattices.

The mapping class group of a surface has various topologically motivated subgroups. In this talk we combine two of them: the Torelli group (of those elements acting trivially on homology) and the handlebody group (of those elements extending to a given handlebody). We prove that it has an (infinite) generating set similar to the usual Torelli group, answering a question of Joan Birman. We also begin to develop a Johnson theory for the handlebody Torelli group, and highlight some of the many open questions about this group. This is joint work-in-progress with Andy Putman.

Over the past few years Church, Ellenberg, Farb, and Nagpal have developed machinery for studying sequences of representations of the symmetric groups, using a concept they call an FI-module. Their work provides a theoretical framework for describing certain stability phenomena for these sequences. I will give an overview of their theory and describe how it generalizes to sequences of representations of the Weyl groups in type B/C and D. I will outline some applications to geometry and topology, including stability results for several families of groups and spaces related to the pure braid groups.

A complex hyperbolic surface is a compact complex manifold X of complex dimension 2 covered by the unit ball B^2 in C^2, that is, X is the quotient of B^2/ by a co-compact, torsion-free lattice in PU(2,1). A closed complex geodesic is a compact (immersed) complex curve that is totally geodesic in the invariant metric. Complex hyperbolic surfaces with arithmetic fundamental group are divided into two classes: those that have no closed complex geodesics, and those that have infinitely many. The purpose of this talk is to prove that, in the latter case, only finitely many closed complex geodesics are embedded. A sharper theorem will be proved: the number of self-intersections grows proportionally to the area. This is in turn a consequence of an equidistribution theorem for complex geodesics derived from Ratner’s theorem. This is joint work with Martin Möller.

We consider an analytic curve $\varphi: I \rightarrow \mathbb{M}(n\times m, \mathbb{R}) \hookrightarrow \mathrm{SL}(n+m, \mathbb{R})$ and embed it into some homogeneous space $G/\Gamma$, and translate it via some diagonal flow $A=\{a(t): t > 0 \} < \mathrm{SL}(n+m,\mathbb{R})$. Under some geometric conditions on $\varphi$, we prove the equidistribution of the evolution of the translated curves $a(t)\varphi(I)$ in $G/\Gamma$, and as a result, we prove that for almost all points on the curve, the Dirichlet's theorem can not be improved. This is a joint work with Nimish Shah.

We will consider a class of groups that includes non-elementary (relatively) hyperbolic groups, mapping class groups, many cubulated groups and C'(1/6) small cancellation groups. Their common feature is to admit an acylindrical action on some Gromov-hyperbolic space and a collection of quasi-geodesics "compatible" with such action. As it turns out, random walks (generated by measures with exponential tail) on such groups tend to stay close to geodesics in the Cayley graph. More precisely, the probability that a given point on a random path is further away than L from a geodesic connecting the endpoints of the path decays exponentially fast in L. This kind of estimate has applications to the rate of escape of random walks (local Lipschitz continuity in the measure) and its variance (linear upper bound in the length). Joint work with Pierre Mathieu.

In 1981 Masur proved the existence of a dense Teichmueller geodesic in moduli space. As some form of analogue, we construct dense geodesic rays in certain subcomplexes of the Out(F_r) quotient of outer space. This is joint work in progress with Yael Algom-Kfir.

The classic conjugacy problem of Max Dehn asks whether, for a given group, there is an algorithm that decides whether pairs of elements are conjugate. Related to this is the following question: given two conjugate elements u,v, what is the shortest length element w such that uw=wv? The conjugacy length function (CLF) formalises this question. I will survey what is known for CLFs of groups, giving a sketch proof for a result in semisimple Lie groups. I will also discuss a new closely related function, the permutation conjugacy length function (PCL), outline its potential application to studying the computational complexity of the conjugacy problem, and describe a result, joint with Y. Antolin, for the PCL of relatively hyperbolic groups.

The Minkowski space R^{2,1} is the Lorentzian analogue of the Euclidean space R^3; the anti-de Sitter space AdS^3 is the Lorentzian analogue of the hyperbolic space H^3. I will survey some recent results on the geometry and topology of their quotients by discrete groups, and explain how the quotients of R^{2,1} by free groups (Margulis spacetimes) are « infinitesimal analogues » of quotients of AdS^3. In particular, we shall see that any Margulis spacetimes admits a fundamental domain bounded by polyhedral surfaces called crooked planes. This is joint work with J. Danciger and F. Guéritaud.

TBA

The action dimension of a group G, actdim(G) is the least dimension of a contractible manifold which admits a proper G-action. The action dimension conjecture states that the L^2-cohomology of any group G vanishes above actdim(G)/2. I will explain the equivalence of this conjecture to the classical Singer conjecture. I will also explain a computation of action dimension for right-angled Artin groups and lattices in Euclidean buildings. This is based on joint work with Grigori Avramidi, Mike Davis, and Boris Okun.

We generalize a problem of Erdos-Szusz-Turan on diophantine approximation to a variety of contexts, and use homogeneous dynamics to compute an associated probability distribution on the integers. This is joint work with Anish Ghosh.

We discuss settings in which interval maps are factors of cross-sections for the geodesic flow on the unit tangent bundle of a hyperbolic surface. That the regular continued fraction, or Gauss, map is of this type was shown by Adler-Flatto and by Series, in the 1980s. With P. Arnoux, we recently affirmatively answered a question of Luzzi-Marmi by showing that the infinite family of so-called Nakada alpha-continued fractions is each of this type. In this talk, we will sketch the background, address the main points of that result, and report on further work.

In many instances one can define the notion of volume of a representation of the fundamental group of a closed manifold M into a simple (non-compact) Lie group G. This is so for instance if M is a surface and the symmetric space associated to G is hermitian, that is carries an invariant 2-form, or if M is a 3-manifold and G is a complex group, equivalently the associated symmetric space carries an invariant 3-form. When M is not compact the definition of volume of a representation presents interesting difficulties; in this talk we will show how bounded cohomology can be used to define an invariant generalizing the volume of a representation and we will see how this invariant is connected with the deformation theory of such representations. This is joint work with Michelle Bucher and Alessandra Iozzi.

In 1985, Boshernitzan showed that a minimal symbolic dynamical system with a linear complexity bound must have a finite number of probability invariant ergodic measures. We will discuss methods to sharpen this bound in general and provide cases in which the bound may already be reduced. This is ongoing work with Michael Damron.

We describe a correspondence between splittings (Bass-Serre decompositions) of a free group of rank n over finitely generated subgroups and systems of surfaces in a doubled handlebody. One can use this to describe a family of hyperbolic complexes on which Out(F_n) acts. This is joint work with Camille Horbez.