NSCPHD1076 Expanders and rigidity of group actions

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Subject area

Expanders are highly connected finite graphs, which arose in work of Barzdin and Kolmogorov from 1967 in connection with understanding the network of nerve cells of the human brain. They have played a fundamental role in computer science (as basic building blocks for various networks and, more recently, in computation theory). The existence of expander graphs follows from probabilistic considerations, but to give explicit constructions, various deep mathematical theories, including Kazhdan's property (T) from representation theory of semi-_‐simple Lie groups (and their discrete subgroups), and the Ramanujan Conjecture from analytic number theory (proved by Deligne), have been used. Building on seminal work of Kazhdan, Margulis gave the first explicit example of expanders, namely, Cayley graphs associated to residually finite property (T) groups, such as SL(3, Z). Over the last 15 years, expanders have provided a wealth of applications to several areas of pure mathematics, including group theory and operator algebras. Here we will address connections to rigidity properties (such as property (T) and fixed point properties) of group actions (by affine isometries) on Banach spaces. Their systematic study (extending classical work of Serre and Delorme‐Guichardet from the Hilbert space setting), was initiated by Monod and collaborators (Invent. Math., 2007). It has revealed important connections to the geometry of the given Banach space.

Scientific  content

The purpose of this PhD Masterclass is to give an introduction to the theory of expander graphs, and  present some of their recent applications to important problems in geometric (and analytic) group theory, and operator algebras. Recent work of de la Salle, and respectively, of Osajda on coarse embeddability of expanders in Banach spaces provided remarkable applications to rigidity properties of group actions on   such spaces, and, respectively, led to the (more accessible) construction of finitely presented groups which are not exact. Until Osajda's results, the only known examples of non-exact groups were the so-called ''Gromov monsters'', whose existence, based on probabilistic methods, was proven by Gromov in 2003, and they are extremely difficult to understand.

The two main series of lectures will be delivered, respectively, by Lubotzky and by Monod. De la Salle and Osajda will give lectures on the applications of embeddability of expanders, as explained above. 

Magdalena Musat is responsible for the overall organization of the Master class, including the lecture series. Christopher Cave is responsible for organizing the problem/discussion sessions. Notes from the lectures will be made available.