On fluctuations of the drift in hyperbolic groups

Abstract

Let $Γ$ refer to a convex-cocompact group of isometries of a CAT($-1$) space $X$ and $Y \to X/Γ$ to a Galois cover with a word-hyperbolic group of deck transformations. We show that, for almost every geodesic $ξ$ with respect to the Bowen-Margulis-Sullivan measure on the lift of $X/Γ$, there exist $\mathfrak{m}, σ> 0$ and a standard Brownian motion $B_s$ such that, for any $λ> 1/4$, \[ d(p(g_s (ξ)), \mathbf{o}) = \mathfrak{m}s + σB_s + o(s^λ), \] with $g_s$ referring to the geodesic flow acting on the geodesics of $Y$ and $p(g_s)$ to the canonical projection to $Y$. The result is a consequence of an almost sure invariance principle for random walks on hyperbolic groups with dependent increments, whose proof makes use of a new Ruelle operator theorem for skew products and Martin boundary techniques for random walks with dependent increments.