Yves Guivarc'h: Random walk by affine maps on tori and nilmanifolds. Spectral gap property in L2

Let X be a compact nilmanifold, T the associated maximal torus factor, p a probability measure supported on Aut(X), P (respQ) the convolution operator defined by p on L2(X), (respL2(T) ). We denote by H the subgroup of Aut(X) generated by supp(p). We show that P has a spectral gap if and only if Q has. We give a necessary and sufficient condition for this to happen:there is no H-equivariant factor torus of T such that the corresponding projection of H is virtually abelian. We discuss a few related situations , in the context of L2-spectral gaps and countable group actions on specific manifolds.