I will give an outline of a proof of the theorem of Masur-Minsky that the curve complex is hyperbolic. The set of nontrivial free factors, up to conjugacy, of a free group of rank n forms the set of vertices of the complex of free factors. I will then discuss hyperbolicity of this complex.

Let \(\Gamma\) be a non-cocompact lattice on a right-angled building \(X\). Examples of such \(X\) include products of trees, or Bourdon's building \(I_{p,q}\), which has apartments hyperbolic planes tesselated by right-angled \(p\)-gons and all vertex links the complete bipartite graph \(K_{q,q}\). We prove that if \(\Gamma\) has a strict fundamental domain then \(\Gamma\) is not finitely generated. The proof uses a topological criterion for finite generation and the separation properties of subcomplexes of \(X\) called tree-walls. This is joint work with Kevin Wortman.

The divergence, \(div(\alpha,r)\), of a geodesic \(\alpha\) measures the length of the shortest path between two points on \(\alpha\) that stays outside the ball of radius \(r\) about their midpoint. We give a group theoretic criterion for determining when a geodesic in a right-angled Artin group \(G\) has super-linear divergence and show that this divergence is at most quadratic. We use this to describe the structure of the asymptotic cone of \(G\) and to get a new proof that every non-abelian subgroup of \(G\) has an infinite dimensional space of quasimorphisms. (Joint work with Jason Behrstock.)

Homological stability is a remarkable phenomenon where for certain sequences \(X_n\) of groups or spaces -- for example \(\mathrm{SL}(n,Z)\), the braid group \(B_n\), or the moduli space \(M_n\) of genus \(n\) curves -- it turns out that the homology groups \(H_i(X_n)\) do not depend on \(n\) once \(n\) is large enough. But for many natural analogous sequences, from pure braid groups to congruence groups to Torelli groups, homological stability fails horribly. In these cases the rank of \(H_i(X_n)\) blows up to infinity, and in the latter two cases almost nothing is known about \(H_i(X_n)\); indeed it's possible there is no nice "closed form" for the answers. While doing some homology computations for the Torelli group, we found what looked like the shadow of an overarching pattern. In order to explain it and to formulate a specific conjecture, we came up with the notion of "representation stability" for a sequence of representations of groups. This makes it possible to meaningfully talk about "the stable homology of the pure braid group" or "the stable homology of the Torelli group" even though the homology never stabilizes. This work is joint with Benson Farb. In this talk I will explain our broad picture and give two major applications. One is a surprisingly strong connection between representation stability for certain configuration spaces and arithmetic statistics for varieties over finite fields, joint with Jordan Ellenberg and Benson Farb. The other is representation stability for the homology of the configuration space of \(n\) distinct points on a manifold \(M\).

Suppose that \(X\) and \(Y\) are surfaces of finite topological type with genus \(g_X\geq 6\) and \(g_Y\leq 2g_X-1\). We describe all homomorphisms \(Map(X)\to Map(Y)\) between the associated mapping class groups. As a consequence we prove that, if \(X\) and \(Y\) have finite analytic type as Riemann surfaces, every non-constant holomorphic map \(M(X)\to M(Y)\) between the corresponding moduli spaces is a forgetful map. This is join work with Javier Aramayona.

For a compact metric space (even as simple as a finite set of points in Euclidean space) there may appear to be nontrivial homology classes at certain fixed scales. I will discuss some recent results on the development of a cohomology theory at a fixed scale for compact metric spaces carrying a Borel measure. For metric spaces that satisfy certain conditions, a corresponding Hodge theory holds. In the special case of Riemannian manifolds at small scales, the corresponding spaces of harmonic functions are isomorphic to the classical ones (and thus the DeRham cohomology).

It is a classical fact, going back to Fricke and Klein, that the relative representation variety of the fundamental group of the quadruply punctured sphere with fixed traces A, B. C. D at the punctures is an affine cubic surface. Since cubic surfaces depend on 4 parameters it is natural question, observed by several authors, whether every cubic surface can be obtained this way. The purpose of this talk is to present a proof of this fact. This is joint work with Bill Goldman.

I will talk about splitting a free group relative to a line pattern.

In this talk I will describe a (contractible) relative outer space on which the group of relative outer automorphisms of a free group acts properly and discontinuously.

I will describe some wild geometry that arises in an apparently benign group theoretic setting: I will exhibit a family of groups enjoying a number of restrictive geometric and algebraic conditions (they are CAT(0), bi-automatic, 1-relator, and free-by-cyclic), and yet these groups have free subgroups of huge (Ackermannian) distortion. The origin of this behaviour lies in a simple computational game --- a realisation of Hercules' battle with the hydra, played out in manipulations of strings of letters. This is work with Will Dison.

A complete Kac-Moody group over a finite field is a totally disconnected, locally compact group, which may be thought of as an "infinite-dimensional Lie group". We study cocompact lattices in such groups of rank 2, where the associated building is a tree, using the group action on the tree and finite group theory. This is joint work with Inna (Korchagina) Capdeboscq.

Bourdon's building is a negatively curved 2-complex built out of hyperbolic right-angled polygons. Its automorphism group is large (uncountable) and remarkably rich. We study, and mostly answer, the question of when there is a discrete subgroup of the automorphism group such that the quotient is a closed surface of genus g. This involves some fun elementary combinatorics, but quickly leads to open questions in group theory and number theory. This is joint work with Anne Thomas.

The first examples of non-arithmetic complex hyperbolic lattices were given by Mostow in 1980. These examples are generalised triangle groups generated by complex reflections of orders 3, 4 or 5. I will discuss how to parametrise such triangle groups and how to identify which of them may possibly be lattices. Most of these candidates are non-arithmetic. I will then survey an ongoing project with Deraux and Paupert whose goal is to use this idea to construct (families of) new non-arithmetic complex hyperbolic lattices.

In order to understand the large scale geometry of limit groups a good starting place is with geometric amalgamations of free groups (a class of graphs of groups) since they are virtually limit groups. In this talk a complete quasi-isometric invariant for geometric amalgamations of free groups will be given, along with an elementary example showing that commensurability and quasi-isometry are not the same equivalence relation for virtual limit groups.

Let S be a closed surface of genus g >= 2. The thick part of the moduli space of S is the set of hyperbolic metrics on S such that the length of the shortest loop is bounded below by a fixed constant. We study the asymptotic behavior of the diameter of this set equipped with the Teichmüller metric and prove that it grows like log(g). This is joint with Kasra Rafi.

abstract

I will prove that every finite volume locally symmetric space contains a compact set K with the property that no periodic maximal flat can be homotoped to be disjoint from K. This is joint work with Juan Souto.

By the work of Morita and Markovic, it is known that the mapping class group of a surface $S$ does not act naturally on $S$. However, such an action, by Hoelder homeomorphisms, exists on the unit tangent bundle $T^1S$ of the surface. In this talk I will explain why this last action is conjugated to a Lipschitz action but not even homotopic to a smooth one.

We show that the extremal length and the hyperbolic length of any simple closed curve are quasi-convex functions of time along any Teichmüller geodesic. As a corollary, we conclude that, in Teichmüller space equipped with the Teichmüller metric, balls are quasi-convex. (Joint work with Anna Lenzhen.)

I'll explain why arithmetic groups of relative Q-type A_n, B_n, C_n, D_n, E_6, and E_7 satisfy an exponential isoperimetric inequality in some dimension.

The Lipschitz metric on Outer Space is not symmetric. In fact d(x,y)/d(y,x) can be arbitrarily large. In joint work with Mladen Bestvina, we define a piecewise differentiable function \psi on Outer Space (which is invariant under the action of Out(Fn) and show that d(x,y) can be bounded in terms of d(y,x) and \psi(x) - \psi(y). I will discuss the proof of this theorem and some applications.

Let f be an isometry between spaces which are products of uniquely geodesic metric spaces with the sup norm. There are two obvious types of isometries from such a space to itself namely a permutation of the factor spaces and a product of isometries of the factor spaces. In this talk we will show that not only is the number of factor spaces an isometry invariant, but also that any isometry is a composition of the two isometries types mentioned above.

I will talk about line patterns in free groups and how they provide quasi-isometry invariants for mapping tori of linearly growing free group automorphisms. This is joint with Natasha Macura.

The first talk will review the concept of asymptotic dimension and some background material. In the second talk I will construct (many) actions of mapping class groups on quasi-trees, and show how this implies that mapping class groups have finite asymptotic dimension. This is joint work with Ken Bromberg and Koji Fujiwara.

The first talk will review the concept of asymptotic dimension and some background material. In the second talk I will construct (many) actions of mapping class groups on quasi-trees, and show how this implies that mapping class groups have finite asymptotic dimension. This is joint work with Ken Bromberg and Koji Fujiwara.

Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| ≤ K (|f| + |g|), where |·| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentially bounded.

Advanced techniques for understanding large scale scientific data are a crucial ingredient in modern science discovery. Developing such techniques involves a number of major challenges in management of massive data, and quantitative analysis of scientific features of unprecedented complexity. Addressing these challenges requires interdisciplinary research in diverse topics including the mathematical foundations of data representations, algorithmic design, and the integration with applications in physics, biology, or medicine. In this talk, I will present a set of case in the use of Morse theory for the representation and analysis of large-scale scientific data. Due to the combinatorial nature of the approach, we can implement the core constructs of Morse theory without the approximations and instabilities of classical numerical techniques. We use topological cancellations to build multi-scale representations that capture local and global trends present in the data. The inherent robustness of our combinatorial algorithms allows us to address the high complexity of the feature extraction problem for high-resolution scientific data.

Picard modular surfaces are the non-compact arithmetic complex hyperbolic 2-orbifolds. I will prove that the two orbifolds studied by John Parker as candidates for orbifolds of smallest volume are indeed the unique arithmetic complex hyperbolic 2-orbifolds of minimal volume. Given time, I will also make some remarks on finding minimal volume manifolds.

The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an open question; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n>=4. This differs from the behavior for n=2 (when the Dehn function is linear) and for n=3 (when it is exponential). In this talk, I will discuss some of the background of the problem and sketch a proof that the Dehn function of SL(n,Z) is at most quartic when n >= 5.

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure M on the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation identities for the Rogers dilogarithm function on the moduli space of a surface.

V has subgroups that are so close to being a BN-pair that the classical proof for simplicity of linear groups with irreducible Coxeter system goes through almost without change. It turns out that the subgroup F plays the role of the solvable Borel subgroup. [joint work with Jim Belk]