# Central Limit Theorems for group actions which are exponentially mixing of all orders

## TL;DR

This paper proves a general dynamical Central Limit Theorem for group actions with exponential mixing of all orders, including applications to Cartan flows and horocycle flows on hyperbolic surfaces.

## Contribution

It introduces a novel relativization of the cumulant method to establish CLTs for exponentially mixing group actions, broadening the scope of dynamical limit theorems.

## Key findings

- CLT holds for Cartan flows on finite-volume quotients of simple Lie groups.
- CLT applies to lacunary samples of horocycle flows on hyperbolic surfaces.
- New proof technique using relativized cumulants enhances understanding of mixing properties.

## Abstract

In this paper we establish a general dynamical Central Limit Theorem (CLT) for group actions which are exponentially mixing of all orders. In particular, the main result applies to Cartan flows on finite-volume quotients of simple Lie groups. Our proof uses a novel relativization of the classical method of cumulants, which should be of independent interest. As a sample application of our techniques, we show that the CLT holds along lacunary samples of the horocycle flow on finite-area hyperbolic surfaces applied to any smooth compactly supported function.