We discuss some general criteria to show that the automorphism group of a homogeneous structure (such as metric spaces, incidence geometries, graphs and hypergraphs) are simple groups or have simple quotients.

Rigidity theorems in geometric group theory prove that a group’s geometry determines its algebra, typically up to virtual isomorphism. We study graphically discrete groups, which impose a discreteness criterion on the automorphism group of any graph the group acts on geometrically. It follows that if a graphically discrete group acts geometrically on the same locally finite graph as another group, then the two are virtually isomorphic. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds; free groups are non-examples. We will present new examples and demonstrate this property is not a quasi-isometry invariant. We will discuss rigidity phenomena for free products of graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

Two flows are said to be Kakutani equivalent if one is isomorphic to the other after time change, or equivalently if there are Poincare sections for the flows so that the respective induced maps are isomorphic to each other. Ratner showed that if G=SL(2,R) and \Gamma is a lattice in G, and if U is a one parameter unipotent subgroup in G then the U action on G equipped with Haar measure is loosely Bernoulli, i.e. Kakutani equivalent to a circle rotation. Thus any two such systems SL(2,R)/\Gamma are Kakutani equivalent to each other. On the other hand, Ratner showed that if G'=SL(2,R) x SL(2,R) and \Gamma' is a reducible lattice, and U' is the diagonally embedded one parameter unipotent subgroup in G', then (G'/\Gamma', U', m) is not loosely Bernoulli. We show that in fact in this case and many other situations one cannot have Kakutani equivalence between such systems unless they are actually isomorphic. This is a joint work with Elon Lindenstrauss.

Surfaces and graphs are closely related; there are many parallels between the mapping class groups of finite-type surfaces and finite graphs, where the mapping class group of a finite graph is the outer automorphism group of a free group of (finite) rank. A recent surge of interest in infinite-type surfaces and their mapping class groups begs a natural question: What is the mapping class group of an “infinite” graph? In this talk, I will explain the answer given by Bestvina and Algom-Kfir, and present recent work, joint with George Domat and Hannah Hoganson, on the coarse geometry of such groups.

The Katok-Spatzier conjecture aims to classify Anosov actions of higher-rank abelian groups. It was originally formulated with the intentionally vague condition of having ``no rank one factors.'' Several recent developments have clarified how rank one factors should be defined, and how they obstruct the smooth classification problem for abelian actions. These include a positive result under stronger asssumptions (joint with R. Spatzier), and a counterexample to the original conjecture, which I will describe. Time permitting I will discuss possible replacements for the conjecture.

It is well-known that the invariant foliations of a pseudo-Anosov surface homeomorphism are transversely orientable if and only if the the pseudo-Anosov dilatation is equal to the spectral radius of its action on homology. This talk will explain a related phenomenon in the context of fully irreducible free group automorphisms. Specifically, we show that equality of geometric and homological stretch factors of the automorphism is a dynamical reflection of the existence of an invariant orientation both on any train track representative and on the expanding lamination of the automorphism. By connecting this to polynomial invariants of free-by-cyclic groups, we are also able to study the extent to which this property of having an orientable monodromy automorphism persists among distinct splittings of a given free-by-cyclic groups. Joint work with Radhika Gupta and Samuel J. Taylor.

We prove an arithmeticity theorem in the context of nonuniform measure rigidity. Adapting machinery developed by Katok and Rodriguez Hertz [J. Mod. Dyn. 10 (2016), 135–172; MR3503686] for Z^k actions to R^k actions, we show that any maximal rank positive entropy action on a manifold generated by k>=2 commuting vector fields of regularity C^r for r>1 is measure theoretically isomorphic to a constant time change of the suspension of some affine action of Z^k on the k-torus or the k-torus modulo {id,-id} with linear parts hyperbolic. Further, the constructed conjugacy has certain smoothness properties. This in particular answers a problem and a conjecture from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin [Ann. of Math. (2) 174 (2011), no. 1, 361–400; MR2811602].

The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.

The horocycle flow has attracted considerable attention in the last century. In the 30’s, Hedlund proved that horocycle orbits are dense in the unit tangent bundles of closed hyperbolic surfaces. For finite volume hyperbolic surfaces, all horocycle orbits are either closed or dense. Our goal is to understand the topology and dynamics of horocycle orbits in geometrically infinite hyperbolic surfaces, where the question is mostly wide open.

In this talk, I will discuss the first complete classification of horocycle orbit closures for a class of Z-covers of closed surfaces. Our analysis is rooted in a seemingly unrelated geometric optimization problem: finding a best Lipschitz map to the circle and identifying the minimizing lamination (!!) of maximal stretch.

This is joint work with Or Landesberg and Yair Minsky.

Motivated by open questions of Sarnak, we explore the relation between algebraic and geometric complexity of simple closed curves on surfaces. We introduce a conjecture on the homological complexity of long simple closed hyperbolic geodesics and proceed to discuss a more accessible problem regarding the action in cohomology of mapping class groups. We explain the relation between these questions and mixing limit theorems for the Kontsevich-Zorich cocycle. We discuss a general framework for upgrading limit theorems to mixing limit theorems for dynamical systems under mild hyperbolicity and ergodicity assumptions. Parts of this talk are joint work in progress with Pouya Honaryar and other parts are joint work in progress with Giovanni Forni.

The fine curve graph is a hyperbolic graph on which the homeomorphism group of a surface acts (in an interesting way). It is motivated by, and shares many properties with, the wildly successful curve graph machinery for mapping class groups — but it also shows new behaviour not encountered in the classical setting. In this talk, we will explore some of this new behaviour by describing (certain) Gromov boundary points and their stabilisers. This is joint work with Jonathan Bowden and Richard Webb

Among the techniques provided by geometric group theory in the study of groups, constructing and exploiting (isometric) actions on hyperbolic spaces and their boundaries is one of the most fruitful and has received a lot of attention in the last decades. In this talk, I focus on groups that are not reachable by such a strategy. I will introduce Property (NL), which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. It turns out that many groups satisfy this property; including many Thompson-like groups. In particular, every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property (NL). I will also talk about the stability of the property under group operations and explore connections to other fixed point properties and the poset of hyperbolic structures. This talk is based on a paper co-authored with F.Fournier and A.Genevois, with an appendix by A.Sisto.

Understanding the geometry of curve graphs is important for proving results on mapping class groups of surfaces. In this talk, we will shed light on the geometry of curve graphs by describing “filtrations” of them by hyperbolic graphs. These graphs are arranged in a sequence via distance non-increasing maps, and the fibers are quasi-trees. This yields a new proof of finite asymptotic dimension of curve graphs. We also calculate the Gromov boundaries and loxodromics of the hyperbolic graphs in the sequence.

Two of the most classical topics of study in geometric group theory are mapping class groups and CAT(0) cube complexes. This is in part because they both admit powerful combinatorial-like structures encoding interesting aspects of their geometries: curve graphs for the former and hyperplanes for the later. The broad class of CAT(0) spaces -- while also studied extensively in the literature-- generally lacks an intrinsic combinatorial structure similar to that present in cube complexes or mapping class groups. I will talk about recent work with Petyt and Spriano where we introduce two combinatorial objects for studying CAT(0) spaces: curtains which are analogues of cubical hyperplanes and the curtain model which is a counter part of the curve graph. Such structures allow for vast extensions of theorems known in the above contexts to that of CAT(0) spaces including an Ivanov-stlyle rigidity theorem, a dichotomy of a rank-rigidity flavor and the presence of a universal hyperbolic space for rank-one elements