Bergelson and Furstenberg have shown that every positive density subset of an amenable, minimally almost periodic group contains a sequence and all its finite products without repetition. In this talk I will present joint work with V. Bergelson, C. Christopherson and P. Zorin-Kranich that strengthens Bergelson and Furstenberg's result and describes a trichotomy of amenable groups relating representation theory with combinatorial properties of positive density subsets.

The main goal of the talk will be to present the following version of the Tits alternative for the outer automorphism group of a free product. Let G_i be freely indecomposable countable groups, and let G=G_1*...*G_k*F_N. If all G_i and Out(G_i) satisfy the Tits alternative, then so does Out(G). Along the proof, I will also discuss classification results for subgroups of Out(F_N). This is partly joint work with Vincent Guirardel.

In this talk I will consider the Teichmuller space of a surface of genus at least 2 equipped with the Teichmuller metric and its quasi-isometry group. Studying the quasi-isometry group of a space has quite a rich history and after introducing the concepts I will review some of this history. Then I will discuss the qi rigidity of Teichmuller space. This is joint work with Alex Eskin and Kasra Rafi.

Commutative algebra can be studied in the general context of a symmetric monoidal abelian category. I will explain some joint work with Andrew Putman and Andrew Snowden in this direction on the category of functors from a category made out of sets or vector spaces to the category of modules over a ring. In some cases, there are strong analogies: a version of the Hilbert basis theorem, rationality of Hilbert series, etc. Time permitting, I will indicate some applications to twisted homological stability for families of groups such as finite linear groups.

To what extent does the algebraic structure of a topological group determine its topology? Many (but not all) examples of real Lie groups G have a unique Lie group structure, meaning that every abstract isomorphism G -> G is necessarily continuous. In this talk, I'll discuss a recent much stronger result for groups of homeomorphisms of manifolds: every homomorphism from Homeo(M) to any other separable topological group is necessarily continuous. This is part of a broader program to show that the topology and algebraic structure of the group of homeomorphisms (or diffeomorphisms) of a manifold M are intimately linked, and also deeply connected to the topology of M itself. Time permitting, we'll discuss applications in geometric topology, groups acting on manifolds, and connections with a new program to study the quasi-isometry type of homeomorphism groups.

The fundamental theorem of projective geometry states that any function that sends points to points and lines to lines and preserves incidence (essentially) comes from a projective transformation. We prove the same theorem in the context of free groups. This is joint work with Martin Bridson.

We introduce a natural chain complex associated to a collection of subspaces of a vector spaces, and discuss the associated homology. We will give some background on Bieri-Neumann-Strebel invariants of groups, and show how the BNS invariant of a group leads to a nice subspace arrangement, whose associated homology is (yet) another invariant of the group. This can give a useful way of distinguishing between finitely presented groups - we will give some examples involving right-angled Artin groups.

We describe a new technique to recast geometric problems involving the 4-ball and ribbon genera of a link in terms of algebraic properties of a braid representative. These techniques make use of braided cobordisms, and require the study of certain shortest word problems in the braid group described by Rudolph. This leads to a new algebraic invariant of the knot. We will present a bound on the solution of this shortest word problem coming from the Dehornoy left-invariant ordering on the braid group.

The Dehn function of a finitely presented group measures the number of relations required to reduce a word which represents the identity to the trivial word. A well-known conjecture states that higher rank S-arithmetic groups have quadratic Dehn functions. An S-arithmetic group sits as a lattice in a product of simple Lie groups. Bestvina-Eskin-Wortman showed that if the G is an S-arithmetic group, and the ambient Lie group containing G has at least 3 factors, then the Dehn function of G is bounded by a polynomial. In this talk, I will use a specific example to illustrate the proof that one can extend the result of Bestvina-Eskin-Wortman to cover certain cases where the ambient Lie group has 2 factors.

Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group had multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in $\mathcal{H}(2)$.

A real reflection in complex hyperbolic space is an antiholomorphic involution (the prototype of such a map is complex conjugation in each coordinate). We provide a concrete criterion to determine whether or not a 2-generator subgroup of SU(2,1) is generated by real reflections. As an application we show that the Picard modular groups SU(2,1,O_d) are generated by real reflections when d=1,2,3,7,11. This is joint work with Pierre Will.

Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation x↦ beta x (mod1), where beta > 1, and replacing some of the branches with branches of constant negative slope. We would like to describe the set of beta for which these maps can admit a Markov partition. We know that beta (which is the exponential of the entropy of the map) must be an algebraic number. Our main result is that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. This extends an analysis of Solomyak for the case of beta-transformations, who obtained a sharp bound of the golden mean in that setting. I will also describe a connection with some of the results of Thurston's fascinating final paper, where the Galois conjugates of entropies of post-critically finite unimodal maps are shown to describe a beautiful fractal. These numbers are included in the setting that we analyze.

The study of the growth of cocycles of compactly generated locally compact groups leads to interesting cohomology theories, e.g. bounded cohomology and polynomially bounded cohomology. In 2006 Monod designated Klingler's cocycle as a good candidate for a `quasification' procedure of Shalom and himself, and asked to determine its growth. This natural cocycle sits in the only non-trivial class of SL_n(F) for the Steinberg representation, where F is a finite extension of the field of p-adic numbers. In this seminar, I will describe the Steinberg representation and Klingler's construction of the cocycle in terms of the Bruhat-Tits building of SL_n(F). Then I will discuss how to compute the norm of the cocycle and present the case of SL_2(F). In rank two, I was lead to study the relative position of three points in the Bruhat-Tits building of SL_3(F).

The classifying space BG of a compact Lie group G (such as a finite group) can be viewed as a direct limit of complex algebraic varieties. As a result, it makes sense to consider the "Chow ring" of algebraic cycles on BG. This ring maps to the cohomology ring of BG, but they are usually not the same. The Chow ring seems to have a close relation to the cobordism of BG. We survey what is known.

There are many contexts where one has a mapping between products, and one would like to know if the product structure must be respected by the map. In geometric group theory, for instance, one would like to know if a quasi-isometry f : H_1 x H_2 ——> G_1 x G_2 between products of groups must asymptotically look like a product of quasi-isometries. The main focus of the lecture will be on the case when the groups are nilpotent, in which case the question is tied with a similar question for bi-Lipschitz homeomorphisms between Carnot groups. It turns out that currents are a key tool for addressing this question. This is joint with Stefan Muller and Xiangdong Xie.

The goal of this talk will be to describe our recent result proving that the asymptotic dimension of the mapping class group of a closed surface is at most quadratic in the genus (building on and strengthening a prior result of Bestvina-Bromberg giving an exponential estimate). We obtain this result as a special case of a result about the asymptotic dimension of a general class of spaces, which we call hierarchically hyperbolic; this class includes hyperbolic spaces, mapping class groups, Teichmueller spaces endowed with either the Teichmuller or the Weil-Petersson metric, fundamental groups of non-geometric 3-manifolds, RAAGs, etc. We will discuss the general framework and a sketch of how this machinery provides new tools for studying special subclasses, such as mapping class groups. The results discussed are joint work with Mark Hagen and Alessandro Sisto.

The study of outer automorphism group of a free group is closely related to the study of mapping class groups of hyperbolic surfaces. In this talk, I will draw analogies between these two groups, and deduce several structural results about subgroups of Out(F) by proving several north-south dynamics type results. Part of this talk is based on joint work with Matt Clay.

Thurston’s skinning map encodes crucial information about the relationship between the asymptotic geometry and the internal geometry of an infinite volume hyperbolic 3-manifold. In the acylindrical case, Thurston’s Bounded Image Theorem asserts that the skinning map has bounded image. This result played a role in Thurston’s original proof of his Geometrization Theorem. In this talk, we will discuss a generalization of Thurston’s result to the setting of deformation spaces of hyperbolic 3-manifolds with freely decomposable fundamental group. The key new tool is a Uniform Core Theorem for sequences of homeomorphic hyperbolic 3-manifolds. We will also discuss potential applications. This is joint work with Ken Bromberg, Jeff Brock and Yair Minsky.

Every holomorphic one-form on a Riemann surface corresponds to a collection of planar polygons with sides identified by translations – a translation surface. The action of GL(2,R) on the plane induces an action on planar polygons and hence on the moduli space of holomorphic one-forms. Work of Eskin, Mirzakhani, Mohammadi, and Filip establishes that GL(2,R) orbit closures are complex subvarieties. We will classify GL(2,R) orbit closures of dimension greater than three in hyperelliptic components of strata and verify a conjecture of Mirzakhani - that higher rank orbit closures arise from loci of branched covers – in these components. As corollaries, we will derive finiteness results for geometrically primitive Teichmuller curves in hyperelliptic components and discuss applications to the illumination and finite blocking problems in rational billiards.

We show that for every nonelementary representation of a surface group into SL(2,C) there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration. Along the way in the argument, we encounter a number of constructions in the geometry of surfaces: complex projective structures, pleated surfaces, harmonic maps to R-trees and the Thurston compactification. Much of the talk will be devoted to explaining the statement and as many of the constructions as time permits. (Joint with Richard Wentworth.)

We construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of exponentially decaying variations. Previously, harmonic maps from the complex plane have been parameterized by holomorphic quadratic differentials. Our harmonic maps, mapping a genus g>1 punctured surface to a k-sided polygon, correspond to meromorphic quadratic differentials with one pole of order (k+2) at the puncture and (4g +k-2) zeros (counting multiplicity) on the Riemann surface domain.

Finite sharply 2- and 3-transitive groups were classified in the 1930 by Zassenhaus and were shown to essentially arise as $AGL_1(K)$ or $PGL_2(K)$ for some field $K$. It remained an open question whether a similar result holds for infinite groups. I will answer this question.

In this talk we discuss the possible closures of geodesic planes in a hyperbolic 3-manifold M. When M has finite volume Shah and Ratner (independently) showed that a very strong rigidity phenomenon holds, and in particular such closures are always properly immersed submanifolds of M with finite area. We show that a similar rigidity phenomenon holds for a class of infinite volume manifolds. The proof uses elements from hyperbolic geometry and Margulis' approach in the proof of the Oppenheim conjecture. This is a joint work with C. McMullen and H. Oh.

We calculate the first homology group of wild topological spaces, like the Hawaiian earring and the harmonic archipelago. To that end, we make use of a result on the combinatorial dynamics of permuting intervals in a linear order, and relate a topological peculiarity to classical notions from abelian group theory.

Work of Avila and Forni established weak mixing for the generic straight line flow on generic translation surfaces, and the work of Avila and Delecroix determined when weak mixing occurs for the straight line flow on a Veech surface. Following the work of Eskin, Mirzakhani, Mohammadi, which proved that the orbit closure of every translation surface has a very nice structure, one can ask how the orbit closure affects the weak mixing of the straight line flow. In this talk all of the necessary background on translation surfaces and weak mixing will be presented followed by the answer to this question. This is a joint work in progress with Artur Avila and Vincent Delecroix.

I will discuss some problems relating to the statistics of the distribution of rational points on homogeneous varieties of semisimple algebraic groups. Joint work with Alexander Gorodnik and Amos Nevo.