We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice Γ in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice.For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Γ, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Γ, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quantitative mean ergodic theorem in L 2 (G/Γ) for a family of averages gives rise to a quantitative solution of the lattice point counting problem in their supports. We demonstrate the new explicit error terms that we obtain by a variety of examples. Contents 1. Main results : Semisimple Lie groups case 3 1.1. Admissible sets 3 1.2. Ergodic theorems on semisimple Lie groups 4 1.3. The lattice point counting problem in admissible domains 6 1.4. Ergodic theorems for lattice subgroups 8 2. Examples and applications 12

Abstract. For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence.The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality.A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ .We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties.We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze-Stein phenomenon. We formulate and prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points.

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The present paper establishes quantitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and thus subject to the generalised Ramanujan conjectures.

Abstract. We consider a finite-dimensional, locally finite CAT(0) cube complex X admitting a co-compact properly discontinuous countable group of automorphisms G. We construct a natural compact metric space B(X) on which G acts by homeomorphisms, the action being minimal and strongly proximal. Furthermore, for any generating probability measure on G, B(X) admits a unique stationary measure, and when the measure has finite logarithmic moment, it constitutes a compact metric model of the Poisson boundary. We identify a dense G δ subset U N T (X) of B(X) on which the action of G is Borel-amenable, and describe the relation of these two spaces to the Roller boundary. Our construction can be used to give a simple geometric proof of Property A for the complex. Our methods are based on direct geometric arguments regarding the asymptotic behavior of half-spaces and their limiting ultrafilters, which are of considerable independent interest. In particular we analyze the notions of median and interval in the complex, and use the latter in the proof that B(X) is the Poisson boundary via the strip criterion developed by V. Kaimanovich [K].