Fixed-point spectrum for group actions by affine isometries on L p -spaces

The fixed-point spectrum of a locally compact second countable group G on ℓ p is defined to be the set p≥1 such that every action by affine isometries admits fixed-point. We show this either empty, or equal one following forms: [1,p c [, [∖{2} for some 1≤p ≤∞, ], ]∖{2} <∞. This result closely related conjecture C. Drutu which asserts connected isometric actions L (0,1).