We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal $p$ for which $L^p$-fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every $p_0 in [2, infty)$ for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on $L^p$-spaces that are $(2-2epsilon)^{1/2p}$-uniformly Lipschitz, and this for every $pin [2,p_0]$. To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemeredis approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results for Kazhdans Property (T) established by Zuk and Ballmann-Swiatkowski.
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