arXiv subjects
Publications and source records attributed to Yves Benoist.
We study the action of the affine group $G$ of $\mathbb{R}^d$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly-supported Zariski dense probability measure $μ$ on $G$, we show that $X_{k,\,d}$ supports a $μ$-stationary measure $ν$ if and only if the $(k\!+\!1)^{\rm th}$-Lyapunov exponent of $μ$ is strictly negative. In particular, when $μ$ is symmetric, $ν$ exists if and only if $2k\geq d$.
We prove a central limit theorem for random walks with finite variance on linear groups.
We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.
We establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple Lie group, generalizing earlier results of Bourgain and Gamburd for unitary groups.
Let G be a semisimple algebraic Lie group and H a reductive subgroup. We find geometrically the best even integer p for which the representation of G in L^2(G/H) is almost L^{p}. As an application, we give a criterion which detects whether this representation is tempered.
Let K be a global field of characteristic not 2. Let Z be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the S-integral points of Z. Our results are effective when K is a number field.
Let G be the group of k-points of a connected reductive k-group and H a symmetric subgroup associated to an involution s of G. We prove a polar decomposition G=KAH for the symmetric space G/H over any local field k of characteristic not 2. Here K is a compact subset of G and A is a finite union of the groups of k-points of maximal (k,s)-split tori, one for each H-conjugacy class. This decomposition is analogous to the well-known polar decomposition G=KAH for a real symmetric space G/H.