Carolyn Abbott - Universal acylindrical actions

The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(F_n) for n > 1. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Given a finitely generated group, one can look for an acylindrical action on a hyperbolic space in which all generalized loxodromic elements act loxodromically; such an action is called a universal acylindrical action. I will discuss recent results in the search for universal acylindrical actions, describing a class of groups for which it is always possible to construct such an action as well as an example of a group for which no such action exists.

David Ayala - Constructing the orthogonal group from finite data.

I'll present a theorem articulating a sense in which the orthogonal group, which consists of linear isometries of Euclidean space, is purely combinatorial, as a topological group. This combinatorial data takes the form of a finite set together with both a group structure on it as well as a partial order on it, with these two structures interacting in a peculiar way. (The statement of this theorem requires a little bit of higher category theory.)

After explaining the statement of this theorem, I'll offer some context for how this result fits into a larger program.

The remainder of the talk will dwell on some key results supporting this theorem. The first is that the Bruhat cells of the orthogonal group form a stratification with adequate regularity to support resolutions of singularity loci. The second is that Bruhat strata multiply in a regular manner.

Talia Fernós - Regular Elements and CAT(0) Cube Complexes

A rank-1 isometry of an irreducible CAT(0) space is an isometry that exhibits hyperbolic-type behavior regardless of whether the ambient space is indeed hyperbolic. A regular isometry of an (essential) CAT(0) cube complex is an isometry that is rank-1 in each irreducible factor. Caprace and Sageev showed that such elements always exist provided that the acting group is nonelementary and a lattice (or just nonelementary in the irreducible case). In a joint work with Lecureux and Matheus, we show that regular elements exist whenever the action is nonelementary. In this talk we will look at examples and discuss key ingredients in the proof of this theorem.

Alex Furman - Asymptotic shapes for ergodic families of metrics on nilpotent groups

In this joint work with Michael Cantrell, we study three closely related problems: Given a finitely generated, virtually nilpotent group $G$ (i) describe the asymptotic cone for an equivariant ergodic family of inner metrics on $G$ (this is a "randomized version" of Pansu's theorem); (ii) describe the limit shapes for First Passage Percolation for general (not necessarily independent) ergodic process on edges of a Cayley graph of $G$; (iii) establish a sub-additive ergodic theorem over a general ergodic G-action. The limiting objects are given in terms of a Carnot-Caratheodory metric on the graded nilpotent group associated to the Mal'cev completion of $G$. In the proof one needs to prove an Ergodic theorem along "polygonal paths" of the form $T_k^n\cdots T_2^nT_1^n$ for elements $T_1,\dots,T_k$ in $G$.

Jingyin Huang - Commensurability of groups quasi-isometric to RAAG's

It is well-known that a finitely generated group quasi-isometric to a free group is commensurable to a free group. We seek higher-dimensional generalization of this fact in the class of right-angled Artin groups (RAAG). Let G be a RAAG with finite outer automorphism group. Suppose in addition that the defining graph of G is star-rigid and has no induced 4-cycle. Then we show every finitely generated group quasi-isometric to G is commensurable to G. However, if the defining graph of G contains an induced 4-cycle, then there always exists a group quasi-isometric to G, but not commensurable to G.

Curtis Kent - Asymptotic cones and boundaries of CAT(0) groups

It is well known that the Tits boundary of a CAT(0) space X embeds in any asymptotic cone of X. This embedding is not preserved under quasi-isometries of the CAT(0) space. We will show how to endow the set of asymptotic cones of a CAT(0) group with a limit structure which respects this canonical embedding of the Tits boundary into the asymptotic cone. We will also show how this limit structure demonstrates the relationship between the visual boundary and the Tits boundary of a CAT(0) group. In particular, we will prove that the direct limit of the asymptotic cones of a CAT(0) space is isometric to the Euclidean cone on its Tits boundary.

Jing Tao - TBA

Dylan Thurston - Energies for maps between graphs

We describe a notion of a "$p$-conformal graph" (essentially a metric graph, but with a particular interpretation of the metric) and energies $E^p_q$ for maps from a $p$-conformal graph to a $q$-conformal graph. Special cases include the length of a curve, Dirichlet (rubber-band) energy, Lipschitz stretching factor, and a versino of extremal length. The energies are sub-multiplicative, in the sense that \[ E^p_r(g \circ f) \le E^p_q(f) E^q_r(g). \] The same inequality is true if we minimize over homotopy classes. Furthermore, these inqualities are all tight in a fairly strong sense. Looking at asymptotic growth of these energies gives a series of invariants $\lambda_q$ associated to virtual endomorphisms of free groups. Among graph maps related to topological branched self-covers of the sphere, the rational maps are those with $\lambda_2 < 1$.