Let $G$ be a reductive group such as $SL_n$ over the field $k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$. The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport. These are indexed by elements $x$ in $G$ and $b$ in $W$, and are related to many important concepts in algebraic geometry over fields of positive characteristic. Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a translation. Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns. Since we work only in the standard apartment of the building for $G$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the p-adics. This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).

# Past Topology Seminar

Building a suitable family of walls in the Cayley complex of a finitely

presented group G leads to a nontrivial action of G on a CAT(0) cube

complex, which shows that G does not have Kazhdan's property (T). I

will discuss how this can be done for certain random groups in Gromov's

density model. Ollivier and Wise (building on earlier work of Wise on

small-cancellation groups) have built suitable walls at densities <1/5,

but their method fails at higher densities. In recent joint work with

Piotr Przytycki we give a new construction which finds walls at densites

<5/24.

I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a

free group $F$ is exactly $Aut(F)$, provided that $F$ has rank at least $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.

The study of closed geodesics on a Riemannian manifold is a classical and important part of differential geometry. In 1969 Gromoll and Meyer used Morse - Bott theory to give a topological condition on the loop space of compact manifold M which ensures that any Riemannian metric on M has an infinite number of closed geodesics. This makes a very close connection between closed geodesics and the topology of loop spaces.

Nowadays it is known that there is a rich algebraic structure associated to the topology of loop spaces — this is the theory of string homology initiated by Chas and Sullivan in 1999. In recent work, in collaboration with John McCleary, we have used the ideas of string homology to give new results on the existence of an infinite number of closed geodesics. I will explain some of the key ideas in our approach to what has come to be known as the closed geodesics problem.

By Thurston's geometrisation theorem, the complement of any knot admits a unique hyperbolic structure, provided that the knot is not the unknot, a torus knot or a satellite knot. However, this is purely an existence result, and does not give any information about important geometric quantities, such as volume, cusp volume or the length and location of short geodesics. In my talk, I will explain how some of this information may be computed easily, in the case of alternating knots. The arguments involve a detailed analysis of the geometry of certain subsurfaces.

It is well known that there are topological obstructions to a manifold $M$ admitting a Riemannian metric of everywhere positive scalar curvature (psc): if $M$ is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of $M$ is invertible, so the vanishing of the $\hat{A}$ genus is a necessary topological condition for such a manifold to admit a psc metric. If $M$ is simply-connected as well as Spin, then deep work of Gromov--Lawson, Schoen--Yau, and Stolz implies that the vanishing of (a small refinement of) the $\hat{A}$ genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research.

I will discuss a related but somewhat different problem: if $M$ does admit a psc metric, what is the topology of the space $\mathcal{R}^+(M)$ of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying $M$ by certain surgeries, and I will explain how this can be used along with work of Galatius and myself to show that the algebraic topology of $\mathcal{R}^+(M)$ for $M$ of dimension at least 6 is "as complicated as can possibly be detected by index-theory". This is joint work with Boris Botvinnik and Johannes Ebert.

I will survey recent advances in our understanding of extended

3-dimensional topological field theories. I will describe recent work (joint

with B. Bartlett, C. Douglas, and J. Vicary) which gives an explicit

"generators and relations" classification of partially extended 3D TFTS

(assigning values only to 3-manifolds, surfaces, and 1-manifolds). This will

be compared to the fully-local case (which has been considered in joint work

with C. Douglas and N. Snyder).

Starting with seminal work by Masur-Minsky, a lot of machinery has been

developed to study the geometry of Mapping Class Groups, and this has

lead, for example, to the proof of quasi-isometric rigidity results.

Parts of this machinery include hyperbolicity of the curve complex, the

distance formula and hierarchy paths.

As it turns out, all this can be transposed to the context of CAT(0)

cube complexes. I will explain some of the key parts of the machinery

and then I will discuss results about quasi-Lipschitz maps from

Euclidean spaces and nilpotent Lie groups into "spaces with a distance

formula".

Joint with Jason Behrstock and Mark Hagen.

Profinite groups are compact totally disconnected groups, or equivalently projective limits of finite groups. This class of groups appears naturally in infinite Galois theory, but they can be studied for their own sake (which will be the case in this talk). We are interested in pro-p groups, i.e. projective limits of finite p-groups. For instance, the group SL(n,Z_p) - and in general any maximal compact subgroup in a Lie group over a local field of residual characteristic p - contains a pro-p group of finite index. The latter groups can be seen as pro-p Sylow subgroups in this situation (they are all conjugate by a non-positive curvature argument).

We will present an a priori non-linear generalization of these examples, arising via automorphism groups of spaces that we will gently introduce: buildings. The main result is the existence of a wide class of automorphism groups of buildings which are simple and whose maximal compact subgroups are virtually finitely generated pro-p groups. This is only the beginning of the study of these groups, where the main questions deal with linearity, and other homology groups.

This is joint work with Inna Cadeboscq (Warwick). We will also discuss related results with I. Capdeboscq and A. Lubotzky on controlling the size of profinite presentations of compact subgroups in some non-Archimedean simple groups