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Boston College
The GEAR Network

HMI

NSF

SFI

# Groups acting on Quasi-Trees

## Schedule of Mini-Course

All minicourse lectures will be held in the Synge Lecture Hall in the Hamilton Building (see campus map).

 Time Speaker Talk Tuesday 9:30 - 10:30 Ken Bromberg (Utah) Groups acting on Quasi-Trees I 10:30 - 11:00 COFFEE & DISCUSSION 11:00 - 12:00 Ken Bromberg (Utah) Groups acting on Quasi-Trees II 12:00 - 2:00 LUNCH 2:00 - 3:00 Richard Webb (Warwick) Tight geodesics in the curve complex 3:00 - 4:00 DISCUSSION Wednesday 9:30 - 10:30 Ken Bromberg (Utah) Groups acting on Quasi-Trees III 10.30 - 11:00 COFFEE & DISCUSSION 11:00 - 12:00 Ken Bromberg (Utah) Groups acting on Quasi-Trees IV 12:00 - 2:00 LUNCH 2:00 - 3:00 Robert Tang (Warwick) On various projection maps in the curve complex 3:00 - 4:00 Tarik Aougab (Yale) Optimal intersection numbers in the curve graph

# Geometry and Groups after Thurston

## Schedule of Talks

All talks will be held in the MacNeil Lecture Hall in the Hamilton Building, TCD (see campus map).

 Time Speaker Talk Thursday 9.00-9.45 Ian Agol (Berkeley) Virtually special cube complexes 9.45-10.15 COFFEE 10:15-11:00 Mladen Bestvina (Utah) Towards the large scale geometry of Out(F_n) 11:15-12:00 Jeff Danciger (Austin) Margulis spacetimes via the arc complex 12:00-2:00 LUNCH 2:00-2:45 Richard Schwartz (Brown) "Turning Squares" 3:00-3:45 Jean-Pierre Otal (Toulouse) Small eigenfunctions on hyperbolic surfaces 4:00-4:45 Richard Canary (Michigan) The pressure metric for convex representations Friday 9:00-9:45 Dave Gabai (Princeton) Topologies of the Curve Complex 9.45-10.15 COFFEE 10:15-11:00 Alan Reid (Austin) Recognizing Surface groups and Kleinian groups by their finite quotients 11:15-12:00 Karen Vogtmann (Cornell) An outer space for right-angled Artin groups 12:00-2:00 LUNCH 2:00-2:45 Jeff Brock (Brown) TBA 3:00-3:45 Genevieve Walsh (Tufts) The Big Dehn surgery graph and the link of S^3 4:00-4:45 Yair Minsky (Yale) Thick-skinned 3-manifolds Saturday 10:00-10:45 Allen Hatcher (Cornell) Configuration Spaces of Branched Polymers 10.45-11.15 COFFEE 11.15-12:00 Dylan Thurston (Indiana) Conformal embeddings of graphs 12.00-2.00 LUNCH 2.00-2.45 Mary Rees (Liverpool) The Resident's View revisited 3.00-3.45 Mahan Mitra (Vivekananda) Homotopical Height END OF WORKSHOP

## Abstracts for Minicourse

Tarik Aougab (Yale)
Title: Optimal intersection numbers in the curve graph
Abstract: Let $S_{g,p}$ be the orientable surface of genus $g$ with $p$ punctures, and let $W(S)= 3g+p-4>0$. Given a natural number $k$, we demonstrate a lower bound on the geometric intersection number for any pair of curves that are a distance of $k$ apart in the corresponding curve graph, and which grows "almost polynomially" of degree $k-2$, in the sense that it grows faster than $(W(S)/2)^{c(k-2)}$, for any $c \in (0,1)$. We use this to prove that curve graphs are uniformly hyperbolic, and that train track splitting sequences project to $R$-quasigeodesics in the curve graph of any essential subsurface, where $R= O(W(S)^{2}). Time permitting, we will demonstrate a construction which shows that these lower bounds on intersection numbers are asymptotically sharp in some suitable sense. Robert Tang (Warwick) Title: On various projection maps in the curve complex Abstract: In my talk, I will discuss some naturally occuring quasiconvex subsets of the curve complex. I will then describe some combinatorial operations, analogous to subsurface projections, which approximate nearest point projection maps to these subsets. Richard Webb (Warwick) Title: Tight geodesics in the curve complex Abstract: Tight geodesics were introduced by Masur and Minsky to develop their hierarchies of curve complexes - these are used in solving Thurston's ending lamination conjecture, as well as having many applications in mapping class groups. Masur and Minsky showed between any pair of vertices in the curve complex there are only finitely many tight geodesics. Bowditch used hyperbolic 3--manifolds to refine further these finiteness properties, in particular showing that the mapping class group acts acylindrically on the curve complex. We shall give a constructive proof of Bowditch's result and some new consequences. We deduce via work of Bell and Fujiwara that curve complexes have asymptotic dimension bounded above by an exponential in Euler characteristic. Therefore by work of Bestvina, Bromberg and Fujiwara, mapping class groups have a computable upper bound on their asymptotic dimension. ## Abstracts for Lecture Series Ian Agol (Berkeley) Title: Virtually special cube complexes Abstract: We'll discuss some of the ingredients of the proof that cubulated hyperbolic groups are virtually special. In particular we'll discuss proofs of Wise's malnormal special quotient theorem and cubulating malnormal amalgams of Hsu and Wise. This is joint work with Daniel Groves and Jason Manning. Mladen Bestvina (Utah) Title: Towards the large scale geometry of Out(F_n) Abstract: I will review the Masur-Minsky theory, which describes the large scale geometry of mapping class groups. Then I will talk about the current progress towards an analogous theory for Out(F_n). Richard Canary (Michigan) Title: The pressure metric for convex representations Abstract: Using the thermodynamics formalism, we introduce a notion of intersection for convex Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out(G)-invariant Riemannian metric on the smooth points of the deformation space of convex, irreducible representations of a word hyperbolic group G into SL(n,R) whose Zariski closure contains a generic element. In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil-Petersson metric on the Fuchsian loci. Moreover, we produce Out(G)-invariant metrics on deformation spaces of convex cocompact representations into PSL(2,C) and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group. This is joint work with Martin Bridgeman, François Labourie and Andres Sambarino. Jeff Danciger (Austin) Title: Margulis spacetimes via the arc complex Abstract: Margulis found the first examples of complete affine three-manifolds with free fundamental group. Associated to each of these manifolds, now called Margulis spacetimes, is a non-compact hyperbolic surface and an infinitesimal deformation of that surface which expands the lengths of closed geodesics in a controlled manner. I will describe a basic construction, called infinitesimal strip deformation, which generates all such expanding surface deformations in an essentially unique way. Roughly, one inserts hyperbolic strips along a collection of disjoint geodesic arcs which fill the surface. This is joint work with François Guéritaud and Fanny Kassel. Dave Gabai (Princeton) Title:Topologies of the Curve Complex Abstract: We compare the topology of the curve complex with its classically defined simplicial structure and its subspace topology induced from PML. For various surfaces, we show how to explicitly construct essential spheres in these spaces. Allen Hatcher (Cornell) Title:Configuration Spaces of Branched Polymers Abstract: This is an unfinished project with Bill Thurston from 2008, with input also from Richard Kenyon. By a branched polymer we mean a finite collection of closed unit disks in the plane such that their union is connected and their interiors are disjoint. Let BP(n) be the space of all such configurations with n disks. One might guess that BP(n) has the same homotopy type as the space of all configurations of n distinct points in the plane, an aspherical space with fundamental group the braid group. However this is false for n at least 8, by a simple argument that Bill found for the case n = 8 that generalizes to larger n as well, showing that BP(n) has a nontrivial homotopy group in dimension 5 when n is 8 or more. This depends strongly on restricting all the disks to have equal radius. Dropping this requirement yields a larger configuration space, where the sizes of all the disks can vary arbitrarily and independently. It appears that this larger space does have the same homotopy type as the space of point configurations, and an argument for this will be sketched, after the argument for the non-asphericity of BP(n) is explained. A weaker conjecture might be that BP(n) at least has fundamental group the braid group, but we were unable to decide this. Mahan Mitra (Vivekananda) Title: Homotopical Height Abstract: We introduce the notion of homotopical height$ht_\CC(G)$of a finitely presented group$G$within a class$\CC$of smooth manifolds with an extra structure (e.g. symplectic, contact, K\"ahler etc). Homotopical height provides an obstruction to finding a$K(G,1)$space within the given class$\CC$. This leads to a hierarchy of these classes in terms of "softness" or "hardness" a la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes$\SP$and$\CA$of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group$G$can be realized as the fundamental group of a manifold in$\SP$and a manifold in$\CA$. For these classes,$ht_\CC(G)$provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of K\"ahler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of K\"ahler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a large class of projective groups (fundamental group of complex projective manifolds) violating property FP. These provide counterexamples to a question of Koll\'ar. This is joint work with Indranil Biswas, Dishant Pancholi Jean Pierre Otal (Toulouse) Title: Small eigenfunctions on hyperbolic surfaces Abstract: The talk will discuss topological properties of Laplace eigenfunctions on negatively curved surfaces, and in particular the theorem that on any hyperbolic surface of genus g, the number of Laplace eigenvalues below 1/4 is at most 2g-2. Mary Rees (Liverpool) Title: The Resident's View revisited Abstract: It is in the nature of Bill Thurston's work that his contributions to all of geometry, topology and dynamics have become foundational in all three areas. It is impossible, for example, to study dynamics in two dimensions without reference to Thurston's isotopy classification of surface homeomorphisms. A related result called Thurston's theorem'' is fundamental in complex dynamics. This result is about postcritically finite branched coverings of the two-sphere, up to the appropriate homotopy equivalence which is always described as Thurston equivalence'', and can be viewed as an analogue, for postcritically finite branched coverings, of the result about surface homeomorphisms. The main technique in Thurston's proof of both results is dynamical, an iteration on a Teichm\"uller space -- and this sort of technique is also crucial in his hyperbolisation theorems. I will talk about work on larger spaces of branched coverings of the two-sphere, defined by relations on the orbits of critical points, and, in particular, on the fundamental groups of such spaces and the actions on the appropriate Teichm\"uller space. This work has been ongoing since 1990 and the main publication was in 2004 (Ast\'erisque 288), but the ideas promoted by Bill Thurston, and our understanding of the geometry of the relevant Teichm\"uller spaces, continue to develop, as I will try to describe. Alan Reid (Austin) Title: Recognizing Surface groups and Kleinian groups by their finite quotients Abstract: This talk will discuss recent work on distinguishing certainclasses of finitely presented groups by their finite quotients. If time permits this will include a surprising application of the work of Agol and Wise. Richard Schwartz (Brown) Title: "Turning Squares" Abstract: I'll describe a group action on the plane where the elements act by infinite rectangle exchanges. The system is based on the idea of turning the squares of a square grid by 90 degrees. These planar systems have higher dimensional compactifications, and in the one case I studied in detail the compactification has a renormalization scheme and is related to the (2,4,infinity) triangle group and to the E4 Weyl group. In my talk I'll show lots of computer pictures and try to survey everything I know about these systems. Dylan Thurston (Indiana) Title: Conformal embeddings of graphs Abstract: Given a branched topological covering$f: (S^2,P) \to (S^2, P)$of the sphere by itself, with branch values contained in$P$, can$f$be realised as a rational map? We give a positive criterion, a counterpart to the obstruction William Thurston found in 1982. We show that f is rational iff there is a metric spine$G$for$S^2\setminus P$so that$f^{-n}(G)$conformally embeds inside$G$for sufficiently large$n$. Here, a map$p: G \to H$between metric graphs is a \emph{conformal embedding} if, for almost all$y \in H$, $\sum_{f(x)=y} |f'(x)| < 1.$ (The intuition is that$G$conformally embeds inside$H$if a slight thickening of$G$conformally embeds as a Riemann surface inside a slight thickening of$H$.) This condition on conformal embeddings of graph relates, in turn, to whether the corresponding virtual endomorphism of$\pi_1(S^2\setminus P)\$ is contracting in a non-standard metric on the group. This is joint-work-in-progress with Kevin Pilgrim.

Karen Vogtmann (Cornell)
Title:  An outer space for right-angled Artin groups
Abstract:  The study of Outer space for a free group has been largely driven by Thurston's study of surface mapping class groups via their action on Teichmuller spaces.  To extend these ideas to rght-angled Artin groups (RAAGs)  one needs a space which can serve the same role, i.e. a contractible space with a proper action of the outer automorphism group.  I will discuss joint work with Ruth Charney in which we construct such a space.

Genevieve Walsh (Tufts)
Title: The Big Dehn surgery graph and the link of S^3.
Abstract: A foundational result of Lickorish and Wallace allows one to create a connected graph where the vertices correspond to closed orientable 3-manifolds and the edges correspond to the existence of a Dehn surgery. This graph was first considered by W. Thurston. We will restate old results and questions in the context of this graph and give some new results and questions related to this graph. For example, we show that there are hyperbolic manifolds with weight one fundamental group which are not obtained via surgery on a knot in S3.   We also explore the geometry of the graph and exhibit some interesting subgraphs.
 Trinity College Dublin, College Green, Dublin 2. Tel: +353-1-608-1000. |