p, li { white-space: pre-wrap; Let G be a locally compact Hausdorff topological group. Examples are Lie groups, p-adic groups, adelic groups, and discrete groups. The BC (Baum-Connes) conjecture proposes an answer to the problem of calculating the K-theory of the convolution C* algebra of G. Validity of the conjecture has implications in several different areas of mathematics --- e.g. Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Dirac exhaustion of the discrete series, Kadison-Kaplansky conjecture. An expander is a sequence of finite graphs which is efficiently connected. Any discrete group which contains an expander as a sub-graph of its Cayley graph is a counter-example to the BC conjecture with coefficients. Such discrete groups have been constructed by Gromov-Arjantseva-Delzant and by Damian Osajda. This talk will indicate how to make a correction in BC with coefficients. There are no known counter-examples to the corrected conjecture, and all previously known confirming examples remain confirming examples.

# Past Topology Seminar

Representations of free loop groups possess an operation, akin to

tensor product, under which they form a braided tensor category. I

will discuss a similar operation, which is present on the category of

representations of the based loop groups, and which equips it with the

structure of a monoidal cateogory. Finally, I will present a recent

result, according to which the Drinfel'd centre of the category of

representations of a based loop group is equivalent to the category of

representations of the corresponding free loop group.

Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then, explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.

The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. At the end I will also discuss a related version of Heegaard Floer homology, which is more computable.

In this talk I will show how given a finitely generated relatively hyperbolic group G, one can construct a finite generating set X of G for which (G,X) has a number of metric properties, provided that the parabolic subgroups have these properties. I will discuss the applications of these properties to the growth series, language of geodesics, biautomatic structures and conjugacy problem. This is joint work with Yago Antolin.

Unordered configuration spaces on (connected) manifolds are basic objects

that appear in connection with many different areas of topology. When the

manifold M is non-compact, a theorem of McDuff and Segal states that these

spaces satisfy a phenomenon known as homological stability: fixing q, the

homology groups H_q(C_k(M)) are eventually independent of k. Here, C_k(M)

denotes the space of k-point configurations and homology is taken with

coefficients in Z. However, this statement is in general false for closed

manifolds M, although some conditional results in this direction are known.

I will explain some recent joint work with Federico Cantero, in which we

extend all the previously known results in this situation. One key idea is

to introduce so-called "replication maps" between configuration spaces,

which in a sense replace the "stabilisation maps" that exist only in the

case of non-compact manifolds. One corollary of our results is to recover a

"homological periodicity" theorem of Nagpal -- taking homology with field

coefficients and fixing q, the sequence of homology groups H_q(C_k(M)) is

eventually periodic in k -- and we obtain a much simpler estimate for the

period. Another result is that homological stability holds with Z[1/2]

coefficients whenever M is odd-dimensional, and in fact we improve this to

stability with Z coefficients for 3- and 7-dimensional manifolds.

If G is a semi-simple Lie group, it is known that all lattices

are arithmetic unless (up to finite index) G=SO(n,1) or SU(n,1).

Non-arithmetic lattices have been constructed in SO(n,1) for

all n and there are infinitely many non-arithmetic lattices in

SU(1,1). Mostow and Deligne-Mostow constructed 9 commensurability

classes of non-arithmetic lattices in SU(2,1) and a single

example in SU(3,1). The problem is open for n at least 4.

I will survey the history of this problem, and then describe

recent joint work with Martin Deraux and Julien Paupert, where

we construct 10 new commensurability classes of non-arithmetic

lattices in SU(2,1). These are the first examples to be constructed

since the work of Deligne and Mostow in 1986.

A permutation group is called sharply n-transitive if it acts freely and transitively on the set of ordered n-tuples of distinct points. The investigation of such permutation groups is a classical branch of group theory; it led Emile Mathieu to the discovery of the smallest finite simple sporadic groups in the 1860's. In this talk I will discuss the case where the permutation group is assumed to be a locally compact transformation group, and explain how this set-up is related to Gromov hyperbolicity and to arithmetic lattices in products of trees.

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).

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.