An important property of Gromov hyperbolic spaces is the fact that every path for which all sufficiently long subpaths are quasi-geodesics is itself a quasi-geodesic. Gromov showed that this property is actually a characterization of hyperbolic spaces. In this talk, we will consider a weakened version of this local-to-global behaviour, called the Morse local-to-global property. The class of spaces that satisfy the Morse local-to-global property include several examples of interest, such as CAT(0) spaces, Mapping Class Groups, fundamental groups of closed 3-manifolds and more. The leverage offered by knowing that a space satisfies this property allows us to import several results and techniques from the theory of hyperbolic groups. In particular, we obtain results relating to stable subgroups, normal subgroups and algorithmic properties.

When Gromov defined non-positively curved cube complexes no one knew what they would be useful for.
Decades latex they played a key role in the resolution of the Virtual Haken conjecture.
In one of the early forays into experimenting with cube complexes, Aitchison, Matsumoto, and Rubinstein produced some nice results about certain "cubed" manifolds, that in retrospect look very prescient.
I will define non-positively curved cube complexes, what it means for a 3-manifold to be cubed, and discuss what all this Haken business is about.
 

We will answer the following question: given a finite simplicial complex X acted on by a finite group G, which object stores the minimal amount of information about the symmetries of X in such a way that we can reconstruct both X and the group action? The natural first guess would be the quotient X/G, which remembers one representative from each orbit. However, it does not tell us the size of each orbit or how to glue together simplices to recover X. Our desired object is, in fact, a complex of groups. We will understand two processes: compression and reconstruction and see primarily through an example how to answer our initial question.

Every Cayley graph of a finitely generated group has some basic properties: they are locally finite, connected, and vertex-transitive. These are not sufficient conditions, there are some well known examples of graphs that have all these properties but are non-Cayley. These examples do however "look like" Cayley graphs, which leads to the natural question of if there exist any vertex-transitive graphs that are completely unlike any Cayley graph. I plan to give some of the history of this question, as well as the construction of the example that finally answered it.