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Register a new userEigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs
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We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general spectral decimation function. This is an extension of earlier results on two dimensional Vicsek sets. Second, we study the Vicsek set lattices, which are natural analogues to the Sierpinski lattices. We have a criterion when two different Vicsek set lattices are isomorphic.
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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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