In this talk I'll give a general presentation of my recent work that a purely loxodromic Kleinian group of Hausdorff dimension<1 is a classical Schottky group. This gives a complete classification of all Kleinian groups of dimension<1. The proof uses my earlier result on the classification of Kleinian groups of sufficiently small Hausdorff dimension. This result in conjunction to another work (joint with Anderson) provides a resolution to Bers uniformization conjecture. No prior knowledge on the subject is assumed.

# Past Topology Seminar

We want to discuss a collection of results around the following Question: Given a smooth compact manifold $M$ without boundary, does $M$ admit a Riemannian metric of positive scalar curvature?

We focus on the case of spin manifolds. The spin structure, together with a chosen Riemannian metric, allows to construct a specific geometric differential operator, called Dirac operator. If the metric has positive scalar curvature, then 0 is not in the spectrum of this operator; this in turn implies that a topological invariant, the index, vanishes.

We use a refined version, acting on sections of a bundle of modules over a $C^*$-algebra; and then the index takes values in the K-theory of this algebra. This index is the image under the Baum-Connes assembly map of a topological object, the K-theoretic fundamental class.

The talk will present results of the following type:

If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has non-trivial index, what conditions imply that $M$ does not admit a metric of positive scalar curvature? How is this related to the Baum-Connes assembly map?

We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$), Engel and new generalizations. Moreover, we will show how these results fit in the context of the Baum-Connes assembly maps for the manifold and the submanifold.

It is known that if the boundary of a 1-ended

hyperbolic group G has a local cut point then G splits over a 2-ended group. We prove a similar theorem for CAT(0)

groups, namely that if a finite set of points separates the boundary of a 1-ended CAT(0) group G

then G splits over a 2-ended group. Along the way we prove two results of independent interest: we show that continua separated

by finite sets of points admit a tree-like decomposition and we show a splitting theorem for nesting actions on R-trees.

This is joint work with Eric Swenson.

We will prove an upper bound for the volume of a minimal

hypersurface in a closed Riemannian manifold conformally equivalent to

a manifold with $Ric > -(n-1)$. In the second part of the talk we will

construct a sweepout of a closed 3-manifold with positive Ricci

curvature by 1-cycles of controlled length and prove an upper bound

for the length of a stationary geodesic net. These are joint works

with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).

Motivated by work of Borel and Serre on arithmetic groups, Bestvina and Feighn defined a bordification of Outer space; this is an enlargement of Outer space which is highly-connected at infinity and on which the action of $Out(F_n)$ extends, with compact quotient. They conclude that $Out(F_n)$ satisfies a type of duality between homology and cohomology. We show that Bestvina and Feighn’s bordification can be realized as a deformation retract of Outer space instead of an extension, answering some questions left open by Bestvina and Feighn and considerably simplifying their proof that the bordification is highly connected at infinity.

Finiteness properties of groups come in many flavours, I will discuss topological finiteness properties. These relate to the finiteness of skelata in a classifying space. Groups with interesting finiteness properties have been found in many ways, however all such examples contains free abelian subgroups of high rank. I will discuss some constructions of groups discussing the various ways we can reduce the rank of a free abelian subgroup.

Bordered Floer homology is a variant of Heegaard Floer homology adapted to manifolds with boundary. I will describe a class of three-manifolds with torus boundary for which these invariants may be recast in terms of immersed curves in a punctured torus. This makes it possible to recast the paring theorem in bordered Floer homology in terms of intersection between curves leading, in turn, to some new observations about Heegaard Floer homology. This is joint work with Jonathan Hanselman and Jake Rasmussen.

For the programme see

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant functions on

the Lie algebra with values in differential forms and a deformation of the de Rham

differential. Before extracting invariants, the Cartan differential does not square

to zero. Unrecognised was the fact that the full complex is a curved algebra,

computing the quotient by G of the algebra of differential forms on X. This

generates, for example, a gauged version of string topology. Another instance of

the construction, applied to deformation quantisation of symplectic manifolds,

gives the BRST construction of the symplectic quotient. Finally, the theory for a

X point with an additional quadratic curving computes the representation category

of the compact group G.