Groups & Dynamics Seminar. Maria Gerasimova (BIU) : “Isoperimetry, Littlewood functions, and unitarisability of group”

Thu, 02/01/202010:00-11:00
Ross 70 a
Let Γ be a discrete group. A group Γ is called unitarisable if for any Hilbert space H and
any uniformly bounded representation π : Γ → B(H) of Γ on H there exists a bounded operator
S : H → H such that S^{−1}π(g)S is a unitary representation for any g ∈ Γ. It is well known that
amenable groups are unitarisable. It has been open ever since whether amenability characterises unitarisability of groups.
Dixmier: Are all unitarisable groups amenable?
One of the approaches to study unitarisability is related to the space of the Littlewood functions
T1(Γ). We define the Littlewood exponent Lit(Γ) of a group Γ:
Lit(Γ) = inf { p : T1(Γ) ⊆ l^p (Γ) }.
We will show that, on the one hand, Lit(Γ) is related to unitarisability and amenability and, on
the other hand, it is related to some geometry of Γ.
We will discuss several applications of this connection. This is a joint work with Dominik Gruber,
Nicolas Monod and Andreas Thom.