The problem of "unbounded rank expanders" asks

whether we can endow a system of generators with a sequence of

special linear groups whose degrees tend to infinity over quotient rings

of Z such that the resulting Cayley graphs form an expander family.

Kassabov answered this question in the affirmative. Furthermore, the

completely satisfactory solution to this question was given by

Ershov and Jaikin--Zapirain (Invent. Math., 2010); they proved

Kazhdan's property (T) for elementary groups over non-commutative

rings. (T) is equivalent to the fixed point property with respect to

actions on Hilbert spaces by isometries.

We provide a new framework to "upgrade" relative fixed point

properties for small subgroups to the fixed point property for the

whole group. It is inspired by work of Shalom (ICM, 2006). Our

main criterion is stated only in terms of intrinsic group structure

(but *without* employing any form of bounded generation).

This, in particular, supplies a simpler (but not quantitative)

alternative proof of the aforementioned result of Ershov and

Jaikin--Zapirain.

If time permits, we will discuss other applications of our result.

# Past Topology Seminar

I will discuss the quantum-field-theory origins of a classic result of Goresky-Kottwitz-MacPherson concerning the Koszul duality between the homology of G and the G-equivariant cohomology of a point. The physical narrative starts from an analysis of supersymmetric quantum mechanics with G symmetry, and leads naturally to a definition of the category of boundary conditions in two-dimensional topological gauge theory, which might be called the "G-equivariant Fukaya category of a point." This simple example illustrates a more general phenomenon (also appearing in C. Teleman's work in recent years) that pure gauge theory in d dimensions seems to control the structure of G-actions in (d-1)-dimensional QFT. This is part of joint work with C. Beem, D. Ben Zvi, M. Bullimore, and A. Neitzke.

My talk is based on joint work with Claire Debord (Univ. Auvergne).

We will explain why Lie groupoids are very naturally linked to Atiyah-Singer index theory.

In our approach -originating from ideas of Connes, various examples of Lie groupoids

- allow to generalize index problems,

- can be used to construct the index of pseudodifferential operators without using the pseudodifferential calculus,

- give rise to proofs of index theorems,

- can be used to construct the pseudodifferential calculus.

When studying a group, it is natural and often useful to try to cut it up

onto simpler pieces. Sometimes this can be done in an entirely canonical

way analogous to the JSJ decomposition of a 3-manifold, in which the

collection of tori along which the manifold is cut is unique up to isotopy.

It is a theorem of Brian Bowditch that if the group acts nicely on a metric

space with a negative curvature property then a canonical decomposition can

be read directly from the large-scale geometry of that space. In this talk

we shall explore an algorithmic consequence of this relationship between

the large-scale geometry of the group and is algebraic decomposition.

I will prove that the knot Floer homology group

HFK-hat(K, g-1) for a genus g fibered knot K is isomorphic to a

variant of the fixed points Floer homology of an area-preserving

representative of its monodromy. This is a joint work with Gilberto

Spano.

(joint work with Françoise Dal'Bo and Andrea Sambusetti)

Given a finitely generated group G acting properly on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. In this work we are interested in the following question: what can we say if H and G have the same exponential growth rate? This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuck and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length). About the same time, Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuck and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory. We focus here one the class of Gromov hyperbolic groups and propose a framework that encompasses both the combinatorial and the geometric point of view. More precisely we prove that if G is a hyperbolic group acting properly co-compactly on a metric space X which is either a Cayley graph of G or a CAT(-1) space, then the growth rate of H and G coincide if and only if H is co-amenable in G. In addition if G has Kazhdan property (T) we prove that there is a gap between the growth rate of G and the one of its infinite index subgroups.

An interval exchange transformation is a map of an

interval to

itself that rearranges a finite number of intervals by translations. They

appear among other places in the

subject of rational billiards and flows of translation surfaces. An

interesting phenomenon is that an IET may have dense orbits that are not

uniformly distributed, a property known as non unique ergodicity. I will

talk about this phenomenon and present some new results about how common

this is. Joint work with Jon Chaika.

A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle. In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, where we prove that the set of all bounce sequences---the "bounce spectrum"---essentially determines the shape of the polygon. This is consequence of a technical result about Liouville currents associated to nonpositively curved Euclidean cone metrics on surfaces. In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of how one determines the shape of the polygon from its bounce spectrum.

It is a truth universally acknowledged, that a local system on a connected topological manifold is completely determined by its attached monodromy representation of the fundamental group. Similarly, lisse ℓ-adic sheaves on a connected variety determine and are determined by representations of the profinite étale fundamental group. Now if one wants to classify constructible sheaves by representations in a similar manner, new invariants arise. In the topological category, this is the exit path category of Robert MacPherson (and its elaborations by David Treumann and Jacob Lurie), and since these paths do not ‘run around once’ but ‘run out’, we coined the term exodromy representation. In the algebraic category, we define a profinite ∞-category – the étale fundamental ∞-category – whose representations determine and are determined by constructible (étale) sheaves. We describe the étale fundamental ∞-category and its connection to ramification theory, and we summarise joint work with Saul Glasman and Peter Haine.