We study random walks on metric spaces with contracting isometries. In this first article of the series, we establish sharp deviation inequalities by adapting Gouëzel’s pivotal time construction. As an application, we establish the exponential bounds for deviation from below, central limit theorem, law of the iterated logarithms, and the geodesic tracking of random walks on mapping class groups and CAT(0) spaces.

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen–Margulis’ measure finiteness assumption used by Ricks [Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. & Dynam. Sys. 37 (2017), 939–970] is removed. We also construct examples of CAT(0) spaces that do not admit finite Bowen–Margulis measure.

In this paper, we investigate a proper $\text{CAT(0)}$ space $(X,\,d)$ that is homeomorphic to ${{\mathbb{R}}^{2}}$ and we show that the asymptotic dimension asdim $(X,\,d)$ is equal to 2.

Given a complete $\text{CAT}(0)$ space $X$ endowed with a geometric action of a group $\Gamma $ , it is known that if $\Gamma $ contains a free abelian group of rank $n$ , then $X$ contains a geometric flat of dimension $n$ . We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the $\text{CAT}(0)$ realization of a Coxeter group $W$ , and $\Gamma $ is a subgroup of $W$ . In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the $\text{CAT}(0)$ realization of arbitrary Tits buildings.