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

The fixed-point spectrum of a locally compact second countable group G on lp is defined to be the set of real numbers p such that every action by affine isometries of G on lp admits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms : [1,\pc[, [1,\pc[\{2} for some \pc<\infty or \pc=\infty, or [1,\pc], [1,\pc]\{2} for some pc<infty. This answers a question closely related to a conjecture of C. Drutu which asserts that the fixed-point spectrum is connected for isometric actions on Lp(0,1). We also study the topological properties of the fixed-point spectrum on Lp(X,μ) for general measure spaces (X,μ), and show partial results toward the conjecture for actions on Lp(0,1). In particular, we prove that the spectrum F_{L^{\infty}(X,μ)(G,π) of actions with linear part πis either empty, or an interval of the form [1,\pc] or [1,\infty[, whenever πis an orthogonal representation associated to a measure-preserving ergodic action on a finite measure space (X,μ).