(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.

# Past Topology Seminar

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.

One of the fundamental themes of geometric group theory is to

view finitely generated groups as geometric objects in their own right,

and to then understand to what extent the geometry of a group determines

its algebra. A theorem of Stallings says that a finitely generated group

has more than one end if and only if it splits over a finite subgroup.

In this talk, I will explain an analogous geometric characterisation of

when a group admits a splitting over certain classes of infinite subgroups.

The Z/2-equivariant Heegaard Floer cohomlogy of the double cover of S^3 along a knot, defined by Lipshitz, Hendricks, and Sarkar,

is an isomorphism class of F_2[\theta]-modules. In this talk, we show that this invariant is natural, and is functorial under based cobordisms.

Given a transverse knot K in the standard contact 3-sphere, we define an element of the Z/2-equivariant Heegaard Floer cohomology

that depends only on the tranverse isotopy class of K, and is functorial under certain symplectic cobordisms.

Rips filtrations over a finite metric space and their corresponding persistent homology are prominent methods in Topological Data Analysis to summarize the ``shape'' of data. For finite metric space $X$ and distance $r$ the traditional Rips complex with parameter $r$ is the flag complex whose vertices are the points in $X$ and whose edges are $\{[x,y]: d(x,y)\leq r\}$. From considering how the homology of these complexes evolves we can create persistence modules (and their associated barcodes and persistence diagrams). Crucial to their use is the stability result that says if $X$ and $Y$ are finite metric space then the bottleneck distance between persistence modules constructed by the Rips filtration is bounded by $2d_{GH}(X,Y)$ (where $d_{GH}$ is the Gromov-Hausdorff distance). Using the asymmetry of the distance function we construct four different constructions analogous to the persistent homology of the Rips filtration and show they also are stable with respect to the Gromov-Hausdorff distance. These different constructions involve ordered-tuple homology, symmetric functions of the distance function, strongly connected components and poset topology.

Developments in geometry and low dimensional topology have given renewed vigour to the following classical question: to what extent do the finite images of a finitely presented group determine the group? I'll survey what we know about this question in the context of 3-manifolds, and I shall present recent joint work with McReynolds, Reid and Spitler showing that the fundamental groups of certain hyperbolic orbifolds are distingusihed from all other finitely generated groups by their finite quotients.

The bipolar filtration of Cochran, Harvey and Horn initiated the study of deeper structures of the smooth concordance group of the topologically slice knots. We show that the graded quotient of the bipolar filtration has infinite rank at each stage greater than one. To detect nontrivial elements in the quotient, the proof uses higher order amenable Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term $d$-invariants simultaneously. This is joint work with Jae Choon Cha.