The ergodic theory of lattice subgroups

TL;DR

This paper develops a comprehensive ergodic theory framework for semisimple algebraic groups and their lattices, providing explicit convergence rates and applications to lattice point counting and equidistribution in various actions.

Contribution

It introduces a general method to derive ergodic theorems for groups and lattices under spectral and geometric conditions, linking ergodic averages to lattice point counting with explicit error bounds.

Findings

Established mean and pointwise ergodic theorems with explicit rates when spectral gaps exist.

Solved lattice point counting problems for general domains in semisimple groups.

Proved equidistribution results for arbitrary isometric actions of lattices.

Abstract

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 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 Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quantitative mean ergodic theorem 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.