Abstract
We prove that a random group in the triangular density model has, for densitylarger than 1/3, fixed point properties for actions on Lpspaces (affineisometric, and more generally (2−2ϵ)1/2puniformly Lipschitz) withp varying in an interval increasing with the set of generators. In the samemodel, we establish a double inequality between the maximal p for whichLpfixed point properties hold and the conformal dimension of the boundary.
In the Gromov density model, we prove that for every p0∈[2,∞)for a sufficiently large number of generators and for any density larger than1/3, a random group satisfies the fixed point property for affine actions onLpspaces that are (2−2ϵ)1/2puniformly Lipschitz, and this forevery p∈[2,p0].
To accomplish these goals we find new bounds on the first eigenvalue of thepLaplacian on random graphs, using methods adapted from Kahn and Szemeredi'sapproach to the 2Laplacian. These in turn lead to fixed point properties usingarguments of Bourdon and Gromov, which extend to Lpspaces previous resultsfor Kazhdan's Property (T) established by Zuk and BallmannSwiatkowski.
In the Gromov density model, we prove that for every p0∈[2,∞)for a sufficiently large number of generators and for any density larger than1/3, a random group satisfies the fixed point property for affine actions onLpspaces that are (2−2ϵ)1/2puniformly Lipschitz, and this forevery p∈[2,p0].
To accomplish these goals we find new bounds on the first eigenvalue of thepLaplacian on random graphs, using methods adapted from Kahn and Szemeredi'sapproach to the 2Laplacian. These in turn lead to fixed point properties usingarguments of Bourdon and Gromov, which extend to Lpspaces previous resultsfor Kazhdan's Property (T) established by Zuk and BallmannSwiatkowski.
Original language  English 

Pages (fromto)  188254 
Number of pages  67 
Journal  Advances in Mathematics 
Volume  341 
Early online date  30 Oct 2018 
DOIs  
Publication status  Published  7 Jan 2019 
Keywords
 Random groups
 Property (T)
 Fixed point properties
 Conformal dimension
 Expanders
 Boundary of hyperbolic groups
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Profiles

Dr John M Mackay
 School of Mathematics  Associate Professor in Pure Mathematics
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member