Geometric Group Theory

- Survey Speakers
- Yair Nathan Minsky (Yale University)
*Structure and deformations of Kleinian groups* - Narutaka Ozawa (Kyoto University)
*Introduction to Noncommutative Real Algebraic Geometry* - Michah Sageev (Technion-Israel Institute of Technology)
*CAT(0) cube complexes and their group actions* - Karen Vogtmann (Cornell University)
*Outer spaces for right-angled Artin groups*

- Yair Nathan Minsky (Yale University)
- Invited Speakers
- Mladen Bestvina (Utah)
*On the asymptotic dimension of a curve complex* - Brian Bowditch (Warwick)
*Quasi-isometric rigidity of mapping class groups and Weil-Petersson geometry* - Martin Bridson (Oxford)
*Cube complexes and profinite isomorphism* - Jeffrey Brock (Brown)
*On the geometry of random hyperbolic 3-manifolds* - Kenneth Bromberg (Utah)
*Bounded cohomology with coefficients in uniformly convex Banach spaces* - Danny Calegari (Chicago)
*Roots, Schottky semigroups, and a proof of Bandt's Conjecture* - David Gabai (Princeton) cancelled
*On the classification of Heegaard Splittings* - Daniel Groves (UIC)
*The Malnormal Special Quotient Theorem* - Yoshikata Kida (Kyoto)
*Stable orbit equivalence relations* - Sang-hyun Kim (Seoul National University)
*RAAGs in Diffeos* - Athanase Papadopoulos (Strasbourg)
*On some theorems in hyperbolic geometry* - Piotr Przytycki (Warsaw)
*Arcs intersecting at most once*

- Mladen Bestvina (Utah)

Title: *Structure and deformations of Kleinian groups*

Abstract: Kleinian groups and hyperbolic 3-manifolds are a surprisingly rich area where ideas from geometry, topology, analysis and dynamics interact in a fruitful way. Structural ideas from this field have inspired much progress in related areas. I will give an introduction and short survey of some of these ideas, focusing on interactions between topological and geometric descriptions of 3-manifolds, and connections to the deformation theory of such structures.

Title: *Introduction to Noncommutative Real Algebraic Geometry*

Abstract: Noncommutative real algebraic geometry is an emerging subject which deals with equations and inequalities in noncommutative algebras such as free polynomial algebras and group algebras. I will overview the general idea of NCRAG and give applications to noncommutative Hilbert 17th problems, Connes's embedding conjecture, and the study of Kazhdan's property (T). No background knowledge is required.

Title: *CAT(0) cube complexes and their group actions*

Abstract: These lectures will be a gentle introduction to the world of CAT(0) cube complexes. CAT(0) cube complexes are non-positively curved spaces which also have a combinatorial structure that turn them into "generalized trees". Besides discussing some basic examples, we will describe the combinatorial and geometric structure of CAT(0) cube complexes via hyperplanes and halfspaces. We plan to explain how groups acting on such spaces often enjoy nice "hyperbolic-like" properties, such as the Tits alternative. We also plan to describe notion of cubulations of groups and briefly describe the connection with subgroup separability and the virtual Haken conjecture.

Title: *Outer spaces for right-angled Artin groups*

Abstract: The first lecture will be a review of the basics of Outer space for a free group: what it is, what it is used for and some recent developments. I will then introduce right-angled Artin groups A_G and their automorphisms, and explain how ideas from the free group case can be modified to build an outer space for the ``untwistedâ€ť subgroup of the outer automorphism group Out(A_G). The third lecture will recall the classical symmetric space for SL(n,Z) and show how to build an outer space for the twist subgroup of Out(A_G). Finally I will discuss how to hybridize the two constructions to obtain an outer space for the entire outer automorphism group Out(A_G). This is joint work with Ruth Charney.

Title: *On the asymptotic dimension of a curve complex*

Abstract: It has been known that the curve complex of a compact surface has finite asymptotic dimension (Bell-Fujiwara) bounded by an exponential function in the complexity of the surface (Webb). In this joint work with K. Bromberg we show that there is a linear bound. The method is to estimate the capacity (or Assouad-Nagata) dimension of the visual boundary by directly constructing required coverings via train tracks on the surface. This uses the work of Buyalo, Gabai and Hamenstadt.

Title: *Quasi-isometric rigidity of mapping class groups and Weil-Petersson geometry*

Abstract: There have been a number of recent results concerning the rigidity of the mapping class groups and related spaces. In particular, the result of Behrstock, Kleiner, Minsky and Mosher, and of Hamenstadt, tells us that any quasi-isometry of the mapping class group is a bounded distance from a map induced by left multiplication. We reinterpret the former argument in terms of coarse medians, and obtain some extensions and variations. For example, one can extend the result to quasi-isometric embeddings. One can also obtain a similar result for quasi-isometries of Teichmuller space in the Weil-Petersson metric for all but at most finitely many surfaces. One can also recover statements about the quasi-isometric rank of such spaces.

Title: *Cube complexes and profinite isomorphism*

Abstract: In this talk I shall outline work with Henry Wilton in which we prove that there is no algorithm that can decide which finitely presented groups have non-trivial finite quotients. A refinement of our construction shows that one cannot decide which compact non-positively curved cube complexes have non-trivial finite-sheeted covers. I shall explain how further refinements can be used to prove that there is no algorithm to decide which finitely presented, residually finite groups have the same set of finite quotients, ie are profinitely isomorphic (and a 'strong genus' variant of this), after presenting the history and context of this problem.

Title: *On the geometry of random hyperbolic 3-manifolds*

Abstract: With respect to various notions of randomness, one can ask what topological and geometric behaviors of a 3-manifold obtain `generically'. In this talk I will discuss a proof of a conjecture of N. Dunfield and W. Thurston that the volume of a random Heegaard splitting grows linearly in the word-length of the gluing map as an element of the mapping class group. In the process, we will see how many other geometric invariants behave in fiberings and splittings arising from random walks in the mapping class group. This talk represents joint work with Igor Rivin and Juan Souto.

Title: *Bounded cohomology with coefficients in uniformly convex Banach spaces*

Abstract: We will show that the 2nd bounded cohomology of acylindrically hyperbolic groups with coefficients in a uniformly convex Banach space is non-trivial. The key case is when the group is F_2, the free group on two generators. This is joint work with M. Bestvina and K. Fujiwara

Title: *Roots, Schottky semigroups, and a proof of Bandt's Conjecture*

Abstract: In 1988, Barnsley and Harrington defined a "Mandelbrot Set" M for pairs of similarities --- this is the set of complex numbers z with |z|<1 for which the limit set of the semigroup generated by the similarities x -> zx and x -> z(x-1)+1 is connected. Equivalently, M is the closure of the set of roots of polynomials with coefficients in {-1,0,1}. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small "holes" in M, and conjectured that these holes were genuine. These holes are very interesting, since they are "exotic" components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, but his methods were not strong enough to show the existence of infinitely many holes; one difficulty with his approach was that he was not able to understand the interior points of M, and on the basis of numerical evidence he conjectured that the interior points are dense away from the real axis. We introduce the technique of *traps* to construct and certify interior points of M, and use them to prove Bandt's Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in M. This is joint work with Sarah Koch and Alden Walker.

Title: *On the classification of Heegaard Splittings*

Abstract: Tao Li proved that a closed non Haken 3-manifold M has only finitely many irreducible Heegaard splittings and gave an algorithm to compute the Heegaard genus of M. It remains to find an algorithm to produce without duplication all the irreducible Heegaard splittings of M. We discuss ongoing efforts towards this end. Joint work with Toby Colding.

Title: *The Malnormal Special Quotient Theorem*

Abstract: The Malnormal Special Quotient Theorem is the key technical result in Wise's work on quasi-convex hierarchies and virtually special cube complexes, and of fundamental importance in Agol's proof of the Virtual Haken Conjecture. I will outline the role of the MSQT in this story, and describe a new proof of (a slight generalization of) the MSQT due to Agol, Manning and myself.

Title: *Stable orbit equivalence relations*

Abstract: A probability-measure-preserving (p.m.p.) action of a discrete countable group associates an orbit equivalence relation on the measure space. If the group is amenable, then the equivalence relation is hyperfinite, that is, approximated by its subrelations all of whose equivalence classes are finite. Hyperfinite equivalence relations are essentially isomorphic to each other, and play a fundamental role in studying equivalence relations. We say that an equivalence relation is stable if it absorbs the hyperfinite equivalence relation under direct products, and say that a group is stable if it admits an ergodic, free and p.m.p. action whose equivalence relation is stable. A question of our interest is which group is stable. Any stable group is necessarily inner amenable. We discuss background of stability and examples of stable groups, including groups having a central subgroup without relative property (T), and the Baumslag-Solitar groups.

Title: *RAAGs in Diffeos*

Abstract: A right-angled Artin group (RAAG) often admits a natural homomorphism into diffeomorphism groups of manifolds, using the fact that two diffeomorphisms with disjoint supports commute. It is a nontrivial task to reinforce this map to become an embedding. We show that an arbitrary RAAG embeds, by a quasi-isometry, into a pure braid group and also into the area-preserving diffeomorphism groups of the disk and of the sphere. We also show that every RAAG embeds into the real line smooth diffeomorphism group. This gives a rich source of embeddings from fundamental groups of manifolds into various diffeomorphism groups. (Joint with Thomas Koberda, partly joint with Hyungryul Baik and Thomas Koberda)

Title: *On some theorems in hyperbolic geometry*

Abstract: I will present some theorems due to Euler and others in spherical geometry and their analogues in hyperbolic geometry. I will discuss a continuous passage from Spherical to hyperbolic geometry. This is based on joint works with Weixu Su and with Norbert A'Campo.

Title: *Arcs intersecting at most once*

Abstract: We prove that on a punctured oriented surface with Euler characteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a closed surface.