John Mackay

Group actions on L^1 spaces

An important way to study groups is through their possible affine
isometric actions on Hilbert spaces. Higher rank lattices, and some
hyperbolic rank 1 groups, have Kazhdan's Property (T): every such action
has a fixed point. But other rank 1 groups such as free groups are
a-T-menable: they admit such an action with a proper orbit. I'll
discuss how these properties behave when "isometric action" is relaxed
to "uniformly Lipschitz action", including a new construction of actions
on $\ell^1$ spaces for groups with hyperbolic features. A key
ingredient is an extension of Brooks' counting quasi-morphisms due to
Bestvina-Bromberg-Fujiwara. Joint work with Cornelia Drutu.