### Group actions on L^1 spaces

Combinatorics Seminar

5th December 2023, 11:00 am – 12:00 pm

Fry Building, 2.04 (NOTE THE CHANGE OF VENUE, BACK TO THE USUAL ONE)

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.

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