We prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus $2$ which are holomorphic limits of discrete series at infinity.

Attached to any reductive Lie group $G$ is a “Cartan motion group” $G_0$—a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_0$, whose existence was suggested by Mackey in the 1970s, has recently been described by the author. In the present article, we use the existence of a family of groups interpolating between $G$ and $G_0$ to realize the bijection as a deformation: for every irreducible tempered representation $\pi$ of G, we build, in an appropriate Fréchet space, a family of subspaces, and evolution operators that contract $\pi$ onto the corresponding representation of $G_0$.

We study random walks on groups, with the feature that, roughly speaking, successive positions of the walk tend to be “aligned.” We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences, including central limit theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In the second part of this article, we show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include nonelementary (relatively) hyperbolic groups, mapping class groups, many groups acting on CAT(0) spaces, and small cancellation groups.