Equidistribution of Dense Subgroups on Nilpotent Lie Groups
Emmanuel Breuillard

TL;DR
This paper proves that dense subgroups of simply connected nilpotent Lie groups become uniformly distributed across the entire group as the word metric radius increases, with explicit rates and random walk analogues.
Contribution
It establishes equidistribution of dense subgroups in nilpotent Lie groups with quantitative rates and extends results to random walk averages.
Findings
$S_n$ becomes equidistributed with Haar measure as n increases.
Provides explicit convergence rates for equidistribution.
Proves a local limit theorem for random walk averages.
Abstract
Let be a dense subgroup of a simply connected nilpotent Lie group generated by a finite symmetric set . We consider the -ball for the word metric induced by on . We show that (with uniform measure) becomes equidistributed on with respect to the Haar measure as n tends to infinity. We give rates and also prove the analogous result for random walk averages (i.e. the local limit theorem).
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Algebra and Geometry · Geometry and complex manifolds
